diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzfojl" "b/data_all_eng_slimpj/shuffled/split2/finalzzfojl" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzfojl" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe standard paradigm in modeling is to postulate that measured quantities contain a contribution of ``accidental deviation'' \\cite{Spearman} from the otherwise ``uniformities'' that characterize an underlying law.\nTherefore, a key issue when identifying dependencies between variables is how to account for the contribution of noise in the data. Various assumptions on the structure of noise and of the possible dependencies lead to a number of corresponding methodologies.\n\nThe purpose of the present paper is to consider from a modern\ncomputational point of view, the important situation where the noise\ncomponents are assumed independent, and the consequences of this\nassumption --the data is typically abstracted into a corresponding\n(estimated) covariance statistic. This independence assumption\nunderlies the errors-in-variables model \\cite{Durbin,KlepperLeamer}\nand factor analysis\n\\cite{AndersonRubin,Ledermann,Harman1966,Joreskog1969,Shapiro}, and\nhas a century-old history \\cite{Frisch2,Reiersol,Koopmans}; see also\n\\cite{Kalman1982,Kalman1985,Los,Woodgate1,Guidorzi95,Soderstrom2007errors,Anderson2008,Forni2000}.\nAccordingly, given the large classical literature on this problem,\nthis paper will also have a tutorial flavor.\n\n\nThe precise formulation has its roots in the work of Ragnar Frisch\nin the 1930's. The central assumption is that the noise components\nare independent of the underlying variables and are also mutually\nindependent \\cite{Kalman1982,Kalman1985}. In addition, since several\nalternative linear relations are typically consistent with the data,\na maximal set of simultaneous dependencies is sought as a means to\nlimit uncertainty and to provide canonical models\n\\cite{Kalman1982,Kalman1985}. This particular dictum gives rise to a\n(non-convex) rank-minimization problem. Thus, it is somewhat\nsurprising that the special case where the maximal number of\npossible simultaneous linear relations is equal to $1$ can be\nexplicitly characterized --this was accomplished over half a\ncentury ago by Reiers{\\o}l \\cite{Reiersol}; see also\n\\cite{Kalman1982,KlepperLeamer}. To date no other case is known that\nadmits a precise closed-form solution.\n\nIn recent years, emphasis has been shifting from hard, non-convex\noptimization to convex regularizations, which in addition scale\nnicely with the size of the problem. Following this trend we revisit\nthe Frisch problem from several alternative angles. We first present\nan overview of the literature, and present several new insights and\nproofs. In the process, we also give an extension of Reiers{\\o}l's\nresult to complex matrices. Our main interest is in exploring\nrecently studied convex optimization problems that approximate rank\nminimization by use of suitable surrogates. In particular, we study\niterative schemes for treating the general Frisch problem and focus\non certificates that guarantee optimality. In parallel, we consider\na viewpoint that serves as an alternative to the Frisch problem\nwhere now, instead of a maximal number of simultaneous linear\nrelations, we seek a uniformly optimal estimator for the unobserved\ndata under the independence assumption of the Frisch scheme. The\noptimal estimator is obtained as a solution to a min-max\noptimization problem. Rank-regularized and min-max alternatives are\ndiscussed and an example is given to highlight the potential and limitations of the techniques.\n\nThe remainder of this paper is organized as follows. We first\nintroduce the errors-in-variables problem in\nSection~\\ref{sec:datastrcuture}. In Section~\\ref{sec:Frisch}, we\nrevisit the Frisch problem, and a related problem due to Shapiro,\nand provide a geometric interpretation of Reiers{\\o}l's result along\nwith a generalization to complex-valued covariances. In\nSection~\\ref{sec:MinTrace}, we present an iterative\ntrace-minimization scheme for solving the Frisch problem and provide\ncomputable lower-bounds for the minimum-rank. In\nSection~\\ref{correspondence}, we bring up the question of estimation\nin the context of the Frisch scheme and motivate a suitable a\nrank-regularized min-max optimization problem in\nSection~\\ref{sec:regularized}. Some concluding remarks are provided\nin Section~\\ref{sec:conclusion}.\n\n\n\\section{Notation}\n\n\n$\\;$\\\\[.1in]\n\\noindent\n\\begin{tabular}{ll}\n ${\\mathcal R}(\\cdot)$, ${\\mathcal N}(\\cdot)$ & range space, null space\\\\\n $\\Pi_{\\mathcal X}$ & orthogonal projection onto ${\\mathcal X}$\\\\\n $>0\\;\\; (\\geq 0)$ & positive definite (resp., positive semi-definite) \\\\\n ${\\mathbf S}_n$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbb R}^{n\\times n},\\; M=M' \\right\\}$\\\\\n ${\\mathbf S}_{n,+}$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbf S}_n,\\; M\\geq0 \\right\\}$\\\\\n ${\\mathbf H}_n$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbb C}^{n\\times n},\\; M=M^* \\right\\}$\\\\\n ${\\mathbf H}_{n,+}$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbf H}_n,\\; M\\geq0 \\right\\}$\\\\\n $[\\cdot ]_{k\\ell},\\;\\; ([\\cdot ]_{k})$ & $(k, \\ell)$-th entry (resp., $k$-th entry)\\\\\n $|M|$& determinant of $M\\in {\\mathbb R}^{n\\times n}$\\\\\n $n_+(\\cdot)$& number of positive eigenvalues\\\\\n ${\\operatorname{diag}}: {\\mathbb R}^{n\\times n} \\to {\\mathbb R}^n: M\\mapsto d$ & where $[d]_i=[M]_{ii}$ for $i=1, \\ldots n$\\\\\n ${\\operatorname{diag}}^*: {\\mathbb R}^{n} \\rightarrow {\\mathbb R}^{n\\times n}: d\\mapsto D$& where $D$ is diagonal and $[D]_{ii}=[d]_{i}$ for $i=1,\\ldots n$\\\\\n $M\\succ_{\\hspace*{-1pt}_e} 0\\;(\\succeq_{\\hspace*{-1pt}_e} 0,\\; \\prec_{\\hspace*{-1pt}_e} 0,\\;\\preceq_{\\hspace*{-1pt}_e} 0)$& the off-diagonal entries are $>0$ (resp.\\ $\\geq 0$, $<0$, $\\leq 0$),\\\\&or \n can be made so by changing the signs of selected\\\\&rows and corresponding columns\\\\\n\\end{tabular}\n\n\\section{Data and basic assumptions}\\label{sec:datastrcuture}\n\nConsider a Gaussian vector ${\\mathbf x}$ taking values in ${\\mathbb R}^{n\\times 1}$ having zero mean and covariance $\\Sigma$. We\nassume that it represents an additive mixture of a Gaussian ``noise-free'' vector ${\\hat{\\mathbf x} }$\nand a ``noise component'' ${\\tilde{\\mathbf x}}$, thus\n\\begin{equation}\\label{eq:xa}\n{\\mathbf x}={\\hat{\\mathbf x} }+{\\tilde{\\mathbf x}}.\n\\end{equation}\nThe entries of ${\\tilde{\\mathbf x}}$ are assumed independent of one another\nand independent of the entries of ${\\hat{\\mathbf x} }$ with both vectors having zero mean\nand covariances $\\hat\\Sigma$ and $\\tilde\\Sigma$, respectively.\nThus,\n\\begin{subequations}\\label{eq:firstsetofconstraints}\n\\begin{eqnarray}\n&&{\\mathcal E}({\\tilde{\\mathbf x}} {\\tilde{\\mathbf x}}') =: \\tilde\\Sigma \\mbox{ is diagonal} \\label{eq:xc}\\\\\n&&{\\mathcal E}({\\hat{\\mathbf x} } {\\tilde{\\mathbf x}}')=0. \\label{eq:xb}\n\\end{eqnarray}\nThroughout ${\\mathcal E}(\\cdot)$ denotes the expectation operation and $0$\ndenotes the zero vector\/matrix of appropriate size. The noise-free\nentries of ${\\hat{\\mathbf x} }$ are assumed to satisfy a set of $q$ simultaneous\nlinear relations. Hence, $M'{\\hat{\\mathbf x} }=0$, with $M\\in {\\mathbb R}^{n\\times q}$ and\n$n>{\\operatorname{rank}}(M)=q>0$. The problem is mainly to infer these relations.\nEquivalently, ${\\mathcal E}({\\hat{\\mathbf x} } {\\hat{\\mathbf x} }') =: \\hat\\Sigma$ has\n\\begin{eqnarray}\n&&{\\operatorname{rank}}(\\hat\\Sigma)= n-q \\label{eq:xd}\n\\end{eqnarray}\n\\end{subequations}\nand $\\hat\\Sigma M=0$. Statistics are typically estimated from\nobservation records. To this end, consider a sequence\n\\[\nx_t\\in{\\mathbb R}^{n\\times 1},\\; t=1,\\ldots,T\n\\]\nof independent measurements (realizations) of ${\\mathbf x}$\nand, likewise, let $\\hat x_t$ and $\\tilde x_t$ represent the corresponding values of the noise-free\nvariable and noise components. Denote by\n\\[\nX=\\left[\\begin{matrix} x_1\\;x_2\\; \\ldots\\; x_T\\end{matrix}\\right]\\in {\\mathbb R}^{n\\times T}\n\\]\nthe matrix of observations of ${\\mathbf x}$ and similarly denote by $\\hat X$\nand $\\tilde X$ the corresponding matrices of the noise-free and\nnoise entries, respectively. Data for identifying relations among\nthe noise-free variables are typically limited to the observation\nmatrix $X$ and, neglecting a scaling factor of $1\/T$, the data is\ntypically abstracted in the form of a sample covariance $XX^\\prime$.\nFor the most part we will assume that sample covariances are\naccurate approximations of true covariances, and hence the modeling\nassumptions amount to\n\\begin{subequations}\n\\begin{eqnarray}\n&& \\tilde X \\tilde X ^\\prime \\simeq \\mbox{ diagonal}\\label{eq:diagonal}\\\\\n&& \\hat X \\tilde X ^\\prime\\simeq 0 \\label{eq:orthogonality}\\\\\n&&{\\operatorname{rank}}(\\hat X) =n-q \\label{eq:rank}\n\\end{eqnarray}\n\\end{subequations}\nsince $M^\\prime \\hat X=0$.\n\nThe number of possible linear relations among the noise free\nvariables and the corresponding coefficient matrix need to be\ndetermined from either $X$ or $\\Sigma$. This motivates the Frisch\nand Shapiro problems discussed in Section~\\ref{sec:Frisch}. An\nalternative set of problems can be motivated by the need to\ndetermine $\\hat X$ from $X$ via suitable decomposition\n\\begin{equation}\\label{eq:decompose}\nX=\\hat X+\\tilde X\n\\end{equation}\nin a way that is consistent with the existence of a set of $q$\nlinear relations. We will return to this in\nSection~\\ref{sec:min-max}.\n\n\n\\section{The problems of Frisch and Shapiro}\\label{sec:Frisch}\n\nWe begin with the Frisch problem concerning the decomposition of a\ncovariance matrix $\\Sigma$ that is consistent with the assumptions\nin Section~\\ref{sec:datastrcuture}. The fact that, in practice,\n$\\Sigma$ is an empirical sample covariance motivates relaxing\n(\\ref{eq:xc}-\\ref{eq:xd}) in various ways. In particular, relaxation\nof the constraint $\\tilde \\Sigma\\geq 0$ leads to the Shapiro\nproblem.\n\n\n\\begin{problem}[\\em The Frisch problem]\\label{problem1} Given $\\Sigma\\in{\\mathbf S}_{n,+}$, determine\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr}}_+(\\Sigma)&:=&\\min\\{{\\operatorname{rank}}(\\hat\\Sigma) \\mid \\Sigma=\\tilde \\Sigma+\\hat \\Sigma,\\\\&&\\tilde\\Sigma, \\hat\\Sigma\\geq 0,\\;\\tilde\\Sigma \\mbox{ is diagonal}\\}.\\label{eq:mc}\n\\end{eqnarray}\n\\end{problem}\n\n\\begin{problem}[\\em The Shapiro problem]\\label{problemShapiro} Given $\\Sigma\\in{\\mathbf S}_{n,+}$, determine\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr}}(\\Sigma)&:=&\\min\\{{\\operatorname{rank}}(\\hat\\Sigma) \\mid \\Sigma=\\tilde \\Sigma+\\hat \\Sigma,\\\\&& \\hat\\Sigma\\geq 0,\\;\\tilde\\Sigma \\mbox{ is diagonal}\\}.\\label{eq:mc2}\n\\end{eqnarray}\n\\end{problem}\n\nThe Frisch problem was studied by several researchers, see e.g.,\n\\cite{Kalman1985,Los,Woodgate1,woodgate2} and the references therein. On the other hand, Shapiro \\cite{Shapiro} introduced the above relaxed\nversion, removing the requirement that $\\tilde \\Sigma\\geq 0$, in an\nattempt to gain understanding of the algebraic constraints imposed\nby the off-diagonal elements of $\\Sigma$ on the decomposition. We\nrefer to ${\\operatorname{mr}}_+(\\cdot)$ as the {\\em Frisch minimum rank} and\n${\\operatorname{mr}}(\\cdot)$ as the {\\em Shapiro minimum rank}. The former is lower\nsemicontinuous whereas the latter is not, as stated next. This\ndifference is crucial if one wants to apply this type of methodology\nto real data, namely some sort of continuity is necessary.\n\n\\begin{prop}\\label{lemma:lowersc}\n${\\operatorname{mr}}_+(\\cdot)$ is lower semicontinuous whereas ${\\operatorname{mr}}(\\cdot)$ is not.\n\\end{prop}\n\n\\begin{proof}\nAssume that for a given $\\Sigma>0$ there exists a sequence $\\Sigma_1,\\,\\Sigma_2,\\,\\ldots$ of positive definite matrices such that\n$\\Sigma_i\\rightarrow \\Sigma$\nwhile\n\\[\n{\\operatorname{mr}}_+(\\Sigma_i)<{\\operatorname{mr}}_+(\\Sigma)=r,\\; \\mbox{ for all }i=1,\\,2,\\,\\dots.\n\\]\nDecompose $\\Sigma_i=\\hat\\Sigma_i+D_i$ with ${\\operatorname{rank}}(\\hat\\Sigma_i)0$.\nClearly ${\\operatorname{mr}}(\\Sigma)=2$. Also $\\lim_{\\epsilon\\to 0}\\Sigma_\\epsilon=\\Sigma$. Yet $\\Sigma_\\epsilon=\n\\hat\\Sigma_\\epsilon+D_\\epsilon$ while $\\Sigma_\\epsilon$\nhas rank $1$ and $D_\\epsilon$ is diagonal ($\\not\\geq 0$). Hence ${\\operatorname{mr}}(\\Sigma_\\epsilon)=1$.\n\\end{proof}\n\nAssuming that the off-diagonal entries of $\\Sigma>0$ of size\n$n\\times n$ are known with absolute certainty, any ``minimum rank''\n(${\\operatorname{mr}}_+(\\cdot)$ and ${\\operatorname{mr}}(\\cdot)$) is bounded below by the so-called\nLederman bound, i.e.,\n\\begin{align}\\label{ledermann}\n\\frac{2n+1-\\sqrt{8n+1}}{2}\\leq {\\operatorname{mr}}(\\Sigma)\\leq {\\operatorname{mr}}_+(\\Sigma),\n\\end{align}\nwhich holds on a generic set of positive definite matrices $\\Sigma$,\nthat is, on a (Zariski open) subset of positive definite matrices.\nEquivalently, the set of matrices $\\Sigma$ for which ${\\operatorname{mr}}(\\Sigma)$\nis lower than the Lederman bound is non-generic --their entries\nsatisfy algebraic equations which fail under small perturbation. To\nsee this, consider any factorization\n\\[\\Sigma =FF^\\prime,\n\\]\nwith $F\\in{\\mathbb R}^{n\\times r}$. There are $(n-r)r + \\frac{r(r+1)}{2}$ independent entries in $F$ (when accounting for the action of a unitary transformation of $F$ on the right), whereas the value of the off-diagonal entries of $\\Sigma$ impose $\\frac{n(n-1)}{2}$ constraints. Thus, the number of independent entries in $F$ exceeds the number of constraints when $(n-r)^2\\geq n+r$ which then leads to the inequality $\\frac{2n+1-\\sqrt{8n+1}}{2}\\leq r$. The bound was first noted in \\cite{Ledermann} while the independence of the constraints has been detailed in \\cite{Bekker1997}.\nIn general, the computation of the exact value for ${\\operatorname{mr}}_+(\\Sigma)$ and ${\\operatorname{mr}}(\\Sigma)$ is a non-trivial matter.\nThus, it is rather surprising that an exact analytic result is available for both, in the special case when $r=n-1$.\nWe review this next in the form of two theorems.\n\n\\begin{thm}[\\em Reiers\\o l's theorem \\cite{Reiersol}] \\label{thm:Reiersol}\nLet $\\Sigma\\in {\\mathbf S}_{n,+}$ and $\\Sigma>0$, then\n\\[\n{\\operatorname{mr}}_+(\\Sigma)=n-1 \\Leftrightarrow \\Sigma^{-1} \\succ_{\\hspace*{-1pt}_e} 0.\n\\]\n\\end{thm}\n\n\n\\begin{thm}[\\em Shapiro's theorem \\cite{Shapiro1982b}]\\label{thm:Shapiro}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ and irreducible,\n\\[\n{\\operatorname{mr}}(\\Sigma)=n-1\\Leftrightarrow \\Sigma\\preceq_{\\hspace*{-1pt}_e} 0.\n\\]\n\\end{thm}\n\nThe characterization of covariance matrices $\\Sigma$ for which\n${\\operatorname{mr}}_+(\\Sigma)=n-1$ was first recognized by T.~C.~Koopmans in 1937\n\\cite{Koopmans} and proven by Reiers\\o l \\cite{Reiersol} who used\nthe Perron-Frobenius theory to improve on Koopmans' analysis. Later\non, R.~E.~Kalman streamlined and completed the steps in\n\\cite{Kalman1982} relying again on the Perron-Frobenius theorem (see\nalso Klepper and Leamer \\cite{KlepperLeamer} for a detailed\nanalysis). Our treatment below takes a slightly different angle and\nprovides some geometric insight by pointing as a key reason that the\nmaximal number of vectors at an obtuse angle from one another can\nexceed the dimension of the ambient space by at most one\n(Corollary~\\ref{cor:numberofobtuseangles}). We provide new proofs\nwhere we also utilize a dual formulation with an analogous\ndecomposition of the inverse covariance.\n\n\\subsection{A geometric insight}\n\nWe begin with two basic lemmas for irreducible matrices in $M\\in{\\mathbf S}_{n,+}$. Recall that a matrix is reducible if by permutation of rows and columns can be brought into a block diagonal form, otherwise it is irreducible.\n\\begin{lemma}\\label{lemma:previous} Let $M>0$ and irreducible. Then,\n\\begin{eqnarray}\\label{eq:first}\nM\\preceq_{\\hspace*{-1pt}_e} 0 &\\Rightarrow & M^{-1}\\succ_{\\hspace*{-1pt}_e} 0.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{lemma}\\label{lemma:next} Let $M\\geq 0$ and irreducible. Then,\n\\begin{eqnarray}\\label{eq:nullitybound}\nM\\preceq_{\\hspace*{-1pt}_e} 0\n&\\Rightarrow & {\\rm nullity}(M)\\leq 1.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nIt is easy to verify that for matrices of size $2\\times 2$,\n(\\ref{eq:first}) holds true. Assume that the statement also holds true for matrices of size up to $k\\times k$, for a certain value of $k$,\nand consider a matrix $M$ of size $(k+1)\\times(k+1)$ with $M>0$ and $M\\preceq_{\\hspace*{-1pt}_e} 0$. Partition\n\\[M=\\left[\\begin{matrix}A &b\\\\b' &c\\end{matrix}\\right]\n\\]\nso that $c$ is a scalar and, hence, $A$ is of size $k\\times k$.\nPartitioning conformably,\n\\[M^{-1}=\\left[\\begin{matrix}F &g\\\\g' &h\\end{matrix}\\right]\n\\]\nwhere\n\\[F=(A-bc^{-1}b')^{-1}, ~g=-A^{-1}bh, \\mbox{ and }h=(c-b'A^{-1}b)^{-1}>0.\n\\]\n\nFor the case where $A$ is irreducible, because $A$ has size $k\\times\nk$ and $A\\preceq_{\\hspace*{-1pt}_e} 0$, invoking our hypothesis we conclude that\n$A^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. Now, since $b$ has only non-positive entries and\n$b\\neq0$, $g=-A^{-1}bh$ has positive entries. Since\n$-bc^{-1}b'\\preceq_{\\hspace*{-1pt}_e} 0$ and $A\\preceq_{\\hspace*{-1pt}_e} 0$, then $A-bc^{-1}b'\\preceq_{\\hspace*{-1pt}_e}\n0$ is also irreducible. Thus $F=(A-bc^{-1}b')^{-1}$ has positive\nentries by hypothesis.\n\nFor the case where $A$ is reducible, permutation of columns and rows\nbrings $A$ into a block-diagonal form with irreducible blocks. Thus,\n$A^{-1}$ is also block diagonal matrix with each block entry-wise\npositive. Because $M$ is irreducible, $b$ must have at least one\nnon-zero entry corresponding to the rows of each diagonal blocks of\n$A$. Then $A-bc^{-1}b'$ is irreducible and $\\preceq_{\\hspace*{-1pt}_e} 0$. Also\n$A^{-1}b$ has all of its entries negative. Therefore\n$F=(A-bc^{-1}b')^{-1}$ and $g=-A^{-1}bh$ have positive entries.\nTherefore $M^{-1}\\succ_{\\hspace*{-1pt}_e} 0$.\n\\end{proof}\n\n\\begin{proof}\nRearrange rows and columns and partition\n\\[M=\\left[\\begin{matrix}A &B\\\\B' &C\\end{matrix}\\right]\n\\]\nso that $A$ is nonsingular and of maximal size, equal to the rank of $M$.\nThen\n\\begin{equation}\\label{eq:equality}\nC=B'A^{-1}B.\n\\end{equation}\n\nWe first show that $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$. Assume that $A$ is irreducible.\nThen $A^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. At the same time $B$ has negative entries and not all zero (since $M$ is irreducible). In this case, $B'A^{-1}B\\succ_{\\hspace*{-1pt}_e} 0$.\nIf on the other hand $A$ is reducible, Lemma \\ref{lemma:previous} applied to the (irreducible) blocks of $A$ implies that $A^{-1}\\succeq_{\\hspace*{-1pt}_e} 0$.\nTherefore, in this case, $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$.\n\nReturning to \\eqref{eq:equality} and in view of the fact that $C\\preceq_{\\hspace*{-1pt}_e} 0$ while $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$ we conclude that, either $C$ is a scalar (and hence there are no off-diagonal negative entries), or both $C$ and $B'A^{-1}B$ are diagonal. The latter contradicts the assumption that $M$ is irreducible. Hence, the nullity of $M$ can be at most $1$.\n\\end{proof}\n\nLemma \\ref{lemma:next} provides the following geometric insight, stated as a corollary.\n\\begin{cor}\\label{cor:numberofobtuseangles} In any Euclidean space of dimension $n$,\nthere can be at most $n+1$ vectors forming an obtuse angle with one another.\n\\end{cor}\n\n\\begin{proof} The Grammian $M=[v_k'v_\\ell]_{k,\\ell=1}^{n+q}$ of a selection $\\{v_k\\mid k=1,\\ldots, n+q\\}$ of such vectors\nhas off-diagonal entries which are negative. Hence, by Lemma \\ref{lemma:next}, the nullity of $M$ cannot exceed $1$.\n\\end{proof}\n\n\nThe necessity part of Theorem \\ref{thm:Shapiro} is also a direct corollary of Lemma \\ref{lemma:next}.\n\\begin{cor}\\label{prop:weaker} Let $\\Sigma\\in{\\mathbf S}_{n,+}$ and irreducible. Then\n\\[\n\\Sigma \\preceq_{\\hspace*{-1pt}_e} 0 \\Rightarrow{\\operatorname{mr}}(\\Sigma)=n-1.\n\\]\n\\end{cor}\n\n\\begin{proof}\nLet\n$\\Sigma =\\hat \\Sigma+\\tilde\\Sigma$, with $\\tilde\\Sigma$ diagonal and $\\hat\\Sigma\\geq 0$. $\\hat\\Sigma$ is irreducible since $\\Sigma$ is irreducible. From Lemma \\ref{lemma:next}, the nullity of $\\hat\\Sigma$ is at most $1$. Thus ${\\operatorname{mr}}(\\Sigma)=n-1$.\n\\end{proof}\n\n\\subsection{A dual decomposition}\n\nThe matrix inversion lemma provides a correspondence between an\nadditive decomposition of a positive-definite matrix and a\ndecomposition of its inverse, albeit with a different sign in one of\nthe summands. This is stated next.\n\n\\begin{lemma}\\label{lemma:decompositions} Let\n\\begin{equation}\\label{eq:first_decomposition}\n\\Sigma=D+FF'\n\\end{equation}\nwith $\\Sigma,D\\in{\\mathbf S}_{n,+}$,\nwith $\\Sigma,D>0$ and $F\\in{\\mathbb R}^{n\\times r}$. Then\n\\begin{equation}\\label{eq:second_decomposition}\nS:=\\Sigma^{-1} = E - GG'\n\\end{equation}\nfor $E=D^{-1}$ and $G=D^{-1}F(I+F'D^{-1}F)^{-1\/2}$. Conversely, if (\\ref{eq:second_decomposition}) holds with $G\\in{\\mathbb R}^{n\\times r}$, then so does (\\ref{eq:first_decomposition}) for $D=E^{-1}$ and $F=E^{-1}G(I-G'E^{-1}G)^{-1\/2}$.\n\\end{lemma}\n\n\\begin{proof} This follows from the identity\n$(I\\pm MM')^{-1}=I\\mp M(I\\mp M'M)^{-1}M'$.\n\\end{proof}\n\n\nApplication of the lemma suggests the following variation to Frisch's problem.\n\\begin{problem}[\\em The dual Frisch problem]\\label{problem2} Given a positive-definite $n\\times n$ symmetric matrix $S$ determine\nthe {\\em dual minimum rank}:\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr_{dual}}}(S)&:=&\\min\\{{\\operatorname{rank}}(\\hat S \\mid S=E -\\hat S,\\\\&& \\hat S,E\\geq 0,\\;E \\mbox{ is diagonal}\\nonumber\\}.\\label{eq:mcdual}\n\\end{eqnarray}\n\\end{problem}\n\nClearly, if $S=\\Sigma^{-1}=E-GG^\\prime$ (as in (\\ref{eq:second_decomposition})), then $E>0$. Furthermore, a\ndecomposition of $S$ always gives rise to\na decomposition $\\Sigma=D+FF^\\prime$ (as in (\\ref{eq:first_decomposition})) with the terms $FF'$ and\n$GG'$ having the same rank. Thus, it is clear that\n\\begin{equation}\\label{eq:inequality}\n{\\operatorname{mr}}_+(\\Sigma)\\leq {\\operatorname{mr_{dual}}}(\\Sigma^{-1}),\n\\end{equation}\nand that the above holds with equality when an optimal choice of $D\\equiv\\tilde \\Sigma$ in (\\ref{eq:mc}) is invertible.\nHowever, if $D$ is allowed to be singular,\nthe rank of the summands $FF'$ and $GG'$ may not agree. This is can be seen using the following example. Take\n\\[\\Sigma=\\left[\\begin{matrix}\n2&1&1\\\\\n1&2&1\\\\\n1&1&1\\end{matrix}\\right].\n\\]\nIt is clear that $\\Sigma$ admits a decomposition\n$\\Sigma=\\tilde\\Sigma+\\hat\\Sigma$, in correspondence with\n(\\ref{eq:first_decomposition}), where\n$\\tilde\\Sigma=D={\\operatorname{diag}}\\{1,1,0\\}$\nwhile $\\hat\\Sigma=FF'$ as well as $F'=[1,\\,1,\\,1]$ are of rank\none. On the other hand,\n\\[\nS=\\Sigma^{-1}=\\left[\\begin{matrix}\n\\;\\;1&\\;\\;0&-1\\\\\n\\;\\;0&\\;\\;1&-1\\\\\n-1&-1&\\;\\;3\\end{matrix}\\right].\n\\]\nTaking $E={\\operatorname{diag}}\\{e_1,\\;e_2,\\;e_3\\}$ in (\\ref{eq:second_decomposition}), it is evident that the rank of\n\\[\nGG'=E-S=\\left[\\begin{matrix}\ne_1-1&0&1\\\\\n0&e_2-1&1\\\\\n1&1&e_3-3\\end{matrix}\\right]\n\\]\ncannot be less than $2$ without violating the non-negativity\nassumption for the summand $GG'$. The minimal rank for the factor\n$G$ is $2$ and is attained by taking $e_1=e_2=2$ and $e_3=5$.\n\nOn the other hand, in general, if we perturb $\\Sigma$ to $\\Sigma+\\epsilon I$ and, accordingly, $D$ to $D+\\epsilon I$, then\n\\begin{equation}\\label{eq:inequality2}\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})\\leq {\\operatorname{mr}}_+(\\Sigma),~ \\forall \\epsilon>0.\n\\end{equation}\nEquality in \\eqref{eq:inequality2} holds for sufficiently small\nvalue of $\\epsilon$. Thus, ${\\operatorname{mr}}_+$ and ${\\operatorname{mr_{dual}}}$ are closely\nrelated. However, it should be noted that ${\\operatorname{mr_{dual}}}(\\cdot)$ fails to\nbe lower semi-continuous since a small perturbation of the\noff-diagonal entries can reduce ${\\operatorname{mr_{dual}}}(\\cdot)$. Yet,\ninterestingly, an exact characterization of the ${\\operatorname{mr_{dual}}}(S)=n-1$ can\nbe obtained which is analogous to those for ${\\operatorname{mr}}_+$ and ${\\operatorname{mr}}$ being\nequal to $n-1$; the condition for ${\\operatorname{mr_{dual}}}$ will be used to prove\nthe Reiers\\o l and Shapiro theorems.\n\n\\begin{thm}\\label{thm:dualreiersol} For $S\\in{\\mathbf S}_{n,+}$, with $S>0$ and irreducible,\n\\begin{equation}\\label{dualreiersol}\n{\\operatorname{mr_{dual}}}(S)=n-1 \\Leftrightarrow S \\succeq_{\\hspace*{-1pt}_e} 0.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nIf $S\\succeq_{\\hspace*{-1pt}_e} 0$ and $E$ is diagonal satisfying $E\\geq S>0$, then $E-S=GG'\\preceq_{\\hspace*{-1pt}_e} 0$.\nBy invoking Lemma~\\ref{lemma:next} we deduce that if $E-S$ is singular, ${\\operatorname{rank}}(G)=n-1$. Hence, ${\\operatorname{mr_{dual}}}(S)=n-1$.\n\nTo establish that ${\\operatorname{mr_{dual}}}(S)=n-1\\Rightarrow S \\succeq_{\\hspace*{-1pt}_e} 0$, we\nassume that the condition $S\\succeq_{\\hspace*{-1pt}_e} 0$ fails and show that\n${\\operatorname{mr_{dual}}}(S)c$ and\n\\[\nM:=E-(A+b(e-c)^{-1}b')\\geq0.\n\\]\nThe nullity of $\\tilde S-S$ coincides with that of $M$. To prove our claim, it suffices to show\nthat $A_e:=A+b(e-c)^{-1}b'\\not\\succeq_{\\hspace*{-1pt}_e} 0$, or that $A_e$ is reducible for some $e>c$. (Since, in either case, by our hypothesis, the nullity of $M$ for a suitable $E$ exceeds $1$.)\n\nWe now consider two possible cases where $S\\succeq_{\\hspace*{-1pt}_e} 0$ fails. First, we consider the case where already $A\\not \\succeq_{\\hspace*{-1pt}_e} 0$.\nThen obviously $A_e\\not\\succeq_{\\hspace*{-1pt}_e} 0$ for $e-c$ sufficiently large.\nThe second possibility is $S\\not \\succeq_{\\hspace*{-1pt}_e} 0$ while $A\\succeq_{\\hspace*{-1pt}_e} 0$. But if $A$ is (transformed into) element-wise nonnegative, then $bb'$ must have at least one pair of negative off-diagonal entries. Then, consider $A_e=A+\\lambda bb'$ for $\\lambda=(e-c)^{-1}\\in(0,\\infty)$. Evidently, for certain values of $\\lambda$ entries of $A_e$ change sign. If a whole row becomes zero for a particular value of $\\lambda$, then $A_e$ is reducible. In all other cases, there are values of $\\lambda$ for which $A_e\\not\\succeq_{\\hspace*{-1pt}_e} 0$. This completes the proof.\n\\end{proof}\n\n\\subsection{Proof of Reiers{\\o}l's theorem (Theorem \\ref{thm:Reiersol})}\\label{sec:proofReiersol}\n\nWe first show that $\\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$ implies\n${\\operatorname{mr}}_+(\\Sigma)=n-1$. From the continuity of the inverse,\n$(\\Sigma+\\epsilon I)^{-1}\\succ_{\\hspace*{-1pt}_e} 0$ for sufficiently small\n$\\epsilon>0$. Applying Theorem~\\ref{thm:dualreiersol}, we conclude\nthat\n\\[\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})=n-1.\n\\]\nSince ${\\operatorname{mr}}_+(\\Sigma)\\geq {\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})$ as in\n\\eqref{eq:inequality2}, we conclude that ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\n\nTo prove that ${\\operatorname{mr}}_+(\\Sigma) =n-1\\Rightarrow \\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$,\nwe show that assuming $\\Sigma^{-1}\\not\\succ_{\\hspace*{-1pt}_e} 0$ and ${\\operatorname{mr}}_+(\\Sigma)\n=n-1$ together leads to a contradiction. From the continuity of the\ninverse and the lower semicontinuity of ${\\operatorname{mr}}_+(\\cdot)$ (Proposition\n\\ref{lemma:lowersc}), there exists a symmetric matrix $\\Delta$ and\nan $\\epsilon>0$ such that\n\\[\n(\\Sigma+\\epsilon \\Delta)^{-1} \\not \\succeq_{\\hspace*{-1pt}_e} 0, \\text{~and~} {\\operatorname{mr}}_+(\\Sigma+\\epsilon \\Delta)=n-1.\n\\]\nThen, from Theorem \\ref{thm:dualreiersol},\n$\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon \\Delta)^{-1})< n-1\n$\nwhile from \\eqref{eq:inequality}\n\\[\n{\\operatorname{mr}}_+(\\Sigma+\\epsilon \\Delta) \\leq {\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon \\Delta)^{-1}).\n\\]\nThus, we have a contradiction and therefore $\\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. $\\Box$\n\n\\subsection{Proof of Shapiro's theorem (Theorem \\ref{thm:Shapiro})}\\label{sec:proofShapiro}\nGiven $\\Sigma\\geq 0$ consider $\\lambda>0$ such that $\\lambda I-\\Sigma\\geq0$, a diagonal $D$, and let $E:=\\lambda I-D$.\nSince\n$\\Sigma-D=E-(\\lambda I -\\Sigma)$,\n\\begin{align}\\label{eq:mrmrdual}\n{\\operatorname{mr}}(\\Sigma)={\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma).\n\\end{align}\nIf $\\Sigma$ is irreducible and $\\Sigma\\preceq_{\\hspace*{-1pt}_e} 0$, then $\\lambda I-\\Sigma$ is irreducible and $\\lambda I-\\Sigma\\succeq_{\\hspace*{-1pt}_e} 0$. It follows (Theorem \\ref{thm:dualreiersol}) that\n${\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma)=n-1$, and therefore ${\\operatorname{mr}}(\\Sigma)=n-1$ as well.\n\nFor the the reverse direction, if ${\\operatorname{mr}}(\\Sigma)=n-1$ then ${\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma)=n-1$, which implies that\n$\\lambda I-\\Sigma\\succeq_{\\hspace*{-1pt}_e} 0$ and therefore that $\\Sigma\\preceq_{\\hspace*{-1pt}_e} 0$. $\\Box$\n\nThe original proof in \\cite{Shapiro1982b} claims that for any $\\Sigma\\geq 0$ of size $n\\times n$ with $n>3$ and $\\Sigma\\not \\preceq_{\\hspace*{-1pt}_e} 0$, there exists a $(n-1)\\times (n-1)$ principle minor that is $\\not\\preceq_{\\hspace*{-1pt}_e} 0$. This statement fails for the following sign pattern\n\\[\\footnotesize{\n\\left[\\begin{matrix}+&0&-&-\\\\0&+&-&+\\\\-&-&+&0\\\\-&+&0&+ \\end{matrix} \\right].}\n\\]\nThis matrix can not transformed to have all nonpositive off-diagonal entries, yet all its $3\\times 3$ principle minors $\\preceq_{\\hspace*{-1pt}_e} 0$.\n\n\n\\subsection{Parametrization of solutions under Reiers{\\o}l's and Shapiro's conditions}\\label{section:parametrization}\n\nFor either the Frisch or the Shapiro problem, a solution is not\nunique in general. The parametrization of solutions to the Frisch\nproblem when ${\\operatorname{mr}}_+(\\Sigma)=n-1$ has been known and is briefly\nexplained below (without proof). Interestingly, an analogous\nparametrization is possible for Shapiro's problem and this is given\nin Proposition~\\ref{shapiro_parametrization} that follows, and both\nare presented here for completeness of the exposition.\n\n\\begin{prop}\\label{lemma:parameter1}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ with $\\Sigma>0$ and $\\Sigma^{-1}\\succe0$. The following hold:\n\\begin{itemize}\n\\item[i)] For $D\\geq0$ diagonal with $\\Sigma-D\\geq0$ and singular, there is a probability vector $\\rho$ ($\\rho$ has entries $\\geq 0$ that sum up to $1$) such that $(\\Sigma-D)\\Sigma^{-1}\\rho=0$.\n\\item[ii)] For any probability vector $\\rho$,\n\\[D={\\operatorname{diag}}^*\\left(\\left[\\frac{[\\rho]_i}{[\\Sigma^{-1}\\rho]_i}, i=1,\\ldots, n\\right] \\right)\n\\]\nsatisfies $\\Sigma-D\\geq0$ and $\\Sigma-D$ is singular.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof} See \\cite{Kalman1982,KlepperLeamer}.\n\\end{proof}\n\nThus, solutions of Frisch's problem under Reiers{\\o}l's conditions\nare in bijective correspondence with probability vectors. A very\nsimilar result holds true for Shapiro's problem.\n\n\\begin{prop}\\label{shapiro_parametrization}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ be irreducible and have $\\leq 0$ off-diagonal entries. The following hold:\n\\begin{itemize}\n\\item[i)]\nFor $D$ diagonal with $\\Sigma-D\\geq0$ and singular, there is a strictly positive vector $v$ such that $(\\Sigma-D)v=0$.\n\\item[ii)] For any strictly positive vector $v\\in {\\mathbb R}^{n\\times 1}$,\n\\begin{align}\\label{eq:Dshapiro}\nD={\\operatorname{diag}}^*\\left(\\left[\\frac{[\\Sigma v]_i}{[v]_i}, i=1,\\ldots, n\\right] \\right)\n\\end{align}\nsatisfies that $\\Sigma-D\\geq0$ and $\\Sigma-D$ is singular.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nTo prove $(i)$, we note that if $(\\Sigma-D)v=0$, then $v\\succ_{\\hspace*{-1pt}_e} 0$.\nTo see this consider $(\\Sigma-D+\\epsilon I)^{-1}$ for $\\epsilon>0$.\nFrom Lemma~\\ref{lemma:previous},\n\\[\n(\\Sigma-D+\\epsilon I)^{-1}\\succ_{\\hspace*{-1pt}_e} 0\n\\]\nand since $v$ is an eigenvector corresponding to its largest eigenvalue, a power iteration argument concludes that $v\\succ_{\\hspace*{-1pt}_e} 0$.\n\nTo prove $ii)$, it is easy to verify that the diagonal matrix $D$ in \\eqref{eq:Dshapiro} for $v\\succ_{\\hspace*{-1pt}_e} 0$\nsatisfies $(\\Sigma-D)v=0$. We only need to prove that $\\Sigma-D\\geq0$. Without loss of generality we assume that all the entries of $v$ are equal.\n(This can always be done by scaling the entries of $v$ and scaling accordingly rows and columns of $\\Sigma$.)\nSince $v$ is a null vector of $\\Sigma-D$ and since $M:=\\Sigma-D$ has $\\leq 0$ off-diagonal entries\n\\[\n[M]_{ii}=\\sum_{j\\neq i}|[M]_{ij}|.\n\\]\nGersgorin Circle Theorem (e.g., see \\cite{Varga2004})\nnow states that every eigenvalue of $M$ lies within at least one of the closed discs $\\left\\{{\\rm Disk}\\left([M]_{ii}, \\sum_{j\\neq i}|[M]_{ij}| \\right), i=1, \\ldots, n\\right\\}$. No disc intersects the negative real line. Therefore $\\Sigma-D\\geq0$.\n\\end{proof}\n\n\\subsection{Decomposition of complex-valued matrices}\n\nComplex-valued covariance matrices are commonly used in radar and\nantenna arrays \\cite{vantrees}. The rank of $\\Sigma-D$, for\nnoise covariance $D$ as in the Frisch problem, is an indication of\nthe number of (dominant) scatterers in the scattering field. If this\nis of the same order as the number of array elements (e.g., $n-1$),\nany conclusion about their location may be suspect. Thus, it is\nnatural to seek conditions for ${\\operatorname{mr}}_+(\\Sigma)=n-1$ analogous to\nthose given by Reiers{\\o}l, for the case of complex covariances, as\na possible warning. This we do next.\n\nConsider complex-valued observation vectors\n$\nx_t=y_t+ {\\rm i} z_t,~ t=1,\\ldots T,\n$\nwhere ${\\rm i}=\\sqrt{-1}$ and $y_t, z_t \\in {\\mathbb R}^{n\\times 1}$, and\nset\n\\[\nX=[x_1,\\; \\ldots x_T]=Y+ {\\rm i} Z\n\\]\nwith\n$Y=[y_1,\\; \\ldots y_T]$,\n$Z=[z_1,\\; \\ldots z_T]$.\nThe (scaled) sample covariance is\n\\begin{align*}\n\\Sigma=XX^*\n&=\\Sigma_{\\rm r}+{\\rm i} \\Sigma_{\\rm i}\\in{\\mathbf H}_{n,+},\n\\end{align*}\nwhere the real part\n$\\Sigma_{\\rm r}:=YY'+ZZ'$ is symmetric,\nthe imaginary part $\\Sigma_{\\rm i}:=ZY'-YZ'$ is anti-symmetric,\nand ``$*$'' denotes complex-conjugate transpose.\nAs before, we consider a decomposition\n\\[\n\\Sigma=\\hat\\Sigma+D\n\\]\nwith $\\hat\\Sigma\\geq 0$ singular and $D\\geq 0$ diagonal.\nWe refer to \\cite{Anderson1988,Deistler1989} for the special case where\n${\\operatorname{mr}}_+(\\Sigma)=1$. In this section we present a sufficient condition for a Reiers{\\o}l-case\nwhere ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\n\nBefore we proceed we note that re-casting the problem in terms\nof the real-valued\n\\[\nR:=\\left[\n \\begin{array}{cc}\n \\Sigma_{\\rm r} & \\Sigma_{\\rm i} \\\\\n \\Sigma_{\\rm i}^\\prime & \\Sigma_{\\rm r} \\\\\n \\end{array}\n \\right]\\in{\\mathbf S}_{2n,+}\n\\]\ndoes not allow taking advantage of earlier results. The structure of $R$ with antisymmetric off-diagonal blocks implies that if $[a',\\;b']'$ is a null vector then so is\n$[-b',\\;a']'$ (since, accordingly, $a+ {\\rm i} b$ and ${\\rm i} a - b$ are both null vectors of $\\Sigma$). Thus, in general, the nullity of $R$ is not $1$ and the theorem of Reiers{\\o}l is not applicable. Further, the corresponding noise covariance is diagonal with repeated blocks.\n\nThe following lemmas for the complex case echo Lemma \\ref{lemma:previous} and Lemma \\ref{lemma:next}.\n\\begin{lemma}\\label{lemma:complexprevious}\nLet $M\\in{\\mathbf H}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n} \\right)$, then\neach entry of $M^{-1}$ has argument in $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right)$.\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to verify the lemma for $2\\times 2$ matrices. Assume that\nthe statement holds for sizes up to $n\\times n$ and consider an\n$(n+1)\\times (n+1)$ matrix $M$ that satisfies the conditions of the\nlemma. Partition\n\\[\nM=\\left[\n \\begin{array}{cc}\n A & b \\\\\n b^* & c \\\\\n \\end{array}\n \\right]\n\\]\nwith $A$ is of size $n\\times n$, and conformably,\n\\[\nM^{-1}=\\left[\n \\begin{array}{cc}\n F & g \\\\\n g^* & h \\\\\n \\end{array}\n \\right].\n\\]\nBy assumption non-zero entries of $-A$ and $-b$ have their argument in $\\left(-\\frac{\\pi}{2^{n+1}}, ~\\frac{\\pi}{2^{n+1}}\\right)$.\nThen, by bounding the possible contribution of the respective terms, it follows that for the argument of each of the entries of $-A+bc^{-1}b^*$ is in\n$\\left(-\\frac{\\pi}{2^n}, ~\\frac{\\pi}{2^n}\\right)$. Then, the argument of each entry of $F=(A-bc^{-1}b^*)^{-1}$ is in\n$\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right)$; this follows by assumption since $F$ is $n\\times n$.\nClearly, $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right) \\subset \\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^{n+1}}, ~\\frac{\\pi}{2}-\\frac{\\pi}{2^{n+1}}\\right)$. Regarding $g$, by bounding the possible contribution of respective terms, we similarly conclude that\nthe argument of each of its non-zero entries is in $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^{n+1}}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^{n+1}}\\right)$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:complexnext}\nLet $M\\in{\\mathbf H}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\\left(-\\frac{\\pi}{2^n},~\\frac{\\pi}{2^n}\\right)$,\nthen ${\\operatorname{rank}}(M)\\geq n-1$.\n\\end{lemma}\n\n\\begin{proof}\nFirst rearrange rows and columns of $M$, and partition as\n\\[\nM=\\left[\n \\begin{array}{cc}\n A & B \\\\\n B^* & C \\\\\n \\end{array}\n\\right]\n\\]\nso that $A$ is nonsingular and of size equal to the rank of $M$, which we denote by $r$. Then\n\\begin{equation}\\label{eq:CBAB}\nC=B^*A^{-1}B\n\\end{equation}\nand has size equal to the nullity of $M$. We now compare the\nargument of the off-diagonal entries of $C$ and $B^*A^{-1}B$, and\nshow they cannot be equal unless $C$ is a scalar. Since the\noff-diagonal entries of $-A$ have their argument in\n$\\left(-\\frac{\\pi}{2^n}, ~\\frac{\\pi}{2^n}\\right)\\subset\n\\left(-\\frac{\\pi}{2^r}, ~\\frac{\\pi}{2^r}\\right)$, the off-diagonal\nentries of $A^{-1}$ have their argument in\n$\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^r}, ~\n\\frac{\\pi}{2}-\\frac{\\pi}{2^r}\\right)$ from Lemma\n\\ref{lemma:complexprevious}. Now, the $(k,\\ell)$ entry of\n$B^*A^{-1}B$ is\n\\begin{align*}\n[B^*A^{-1}B]_{k\\ell}=\\sum_{i,j}[B^*]_{ki}[A^{-1}]_{ij}[B]_{j \\ell}\n\\end{align*}\nand the phase of each summand is\n\\[\n\\arg([B^*]_{ki}[A^{-1}]_{ij} [B]_{j \\ell}) \\in\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^r}-\\frac{\\pi}{2^{n-1}},~ \\frac{\\pi}{2}-\\frac{\\pi}{2^r}+\\frac{\\pi}{2^{n-1}}\\right).\n\\]\nThus, the non-zero off-diagonal entries of $B^*A^{-1}B$ have positive real part while\n \\[\n \\arg(-[C]_{k\\ell})\\in \\left(-\\frac{\\pi}{2^n},~\\frac{\\pi}{2^n}\\right) .\n \\]\nHence, either the off-diagonal entries of $B^*A^{-1}B$ and $C$ are\nzero, in which case these are diagonal matrices and $M$ must be\nreducible, or $B^*A^{-1}B$ and $C$ are both scalars. This concludes\nthe proof.\n\\end{proof}\n\n\\begin{thm}\\label{prop:complexReiersol}\nLet $\\Sigma\\in{\\mathbf H}_{n,+}$ be irreducible.\nIf the argument of each non-zero off-diagonal entry of $-\\Sigma$ is in\n$\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, then\n${\\operatorname{mr}}(\\Sigma)=n-1$.\n\\end{thm}\n\n\\begin{proof}\nThe matrix $\\Sigma-D$ is irreducible since $D$ is diagonal.\nIf $\\Sigma-D\\geq0$ and singular, and since the argument of each non-zero off-diagonal entry of $-(\\Sigma-D)$ is in\n$\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, Lemma \\ref{lemma:complexnext} applies and gives that ${\\operatorname{rank}}(\\Sigma-D)=n-1$.\n\\end{proof}\n\nClearly, since ${\\operatorname{mr}}_+(\\Sigma)\\geq{\\operatorname{mr}}(\\Sigma)$, under the condition of Theorem \\ref{prop:complexReiersol}, ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\nIt is also clear that for $S\\in{\\mathbf H}_{n,+}$ irreducible with all non-zero off-diagonal entries having argument in $\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, we also conclude that\n${\\operatorname{mr_{dual}}}(S)=n-1$.\n\n\\section{Trace minimization heuristics}\\label{sec:MinTrace}\n\nThe rank of a matrix is a non-convex function of its elements and the problem to find the matrix of minimal rank within a given set is a difficult one, in general.\nTherefore, certain heuristics have been developed over the years to obtain approximate solutions.\nIn particular, in the context of factor analysis, trace minimization has been pursued as a suitable heuristic \\cite{Ledermann1940,Shapiro,Shapiro1982b} thereby relaxing the Frisch problem into\n\\begin{align}\\nonumber\n&\\min_{D: \\Sigma\\geq D\\geq0} {\\operatorname{trace}}(\\Sigma-D),\n\\end{align}\nfor a diagonal matrix $D$; with a relaxation of $D\\geq 0$ corresponding to Shapiro's problem. The theoretical basis for using the trace and, more generally, the nuclear norm for non-symmetric matrices, as a surrogate for the rank was provided by\nFazel {\\em etal.} \\cite{Fazel2001} who proved that these constitute convex envelops of the rank function on bounded sets of matrices.\n\nThe relation between minimum trace factor analysis and minimum rank factor analysis goes back to Ledermann in \\cite{Ledermann1939} (see \\cite{Della1982} and \\cite{Saunderson2012}). Herein we only refer to two propositions which characterize minimizers for the two problems, Frisch's and Shapiro's, respectively.\n\n\\begin{subequations}\n\\begin{prop}[\\cite{Della1982}]\\label{prop:mintrace}\nLet $\\Sigma=\\hat\\Sigma_1+D_1>0$ for a diagonal $D_1\\geq0$. Then,\n\\begin{align}\\label{trmc}\n &(\\hat\\Sigma_1,D_1)=\\arg\\min\\{ {\\operatorname{trace}}(\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq 0,\\;\\mbox{diagonal }D\\geq 0\\}\\\\\n& \\Leftrightarrow~ \\exists~ \\Lambda_1 \\geq0 ~:~ \\hat\\Sigma_1 \\Lambda_1=0 \\text{~and~} \\left\\{\n \\begin{array}{ll}\n [\\Lambda_1]_{ii}=1, & \\text{~if~} [D_1]_{ii}>0, \\\\\n \\left[ \\Lambda_1\\right]_{ii}\\geq1, &\\text{~if~}[D_1]_{ii}=0.\\nonumber\n \\end{array}\n \\right.\n\\end{align}\n\\end{prop}\n\n\\begin{prop}[\\cite{Saunderson2012}]\\label{prop:MTFAShapiro}\nLet $\\Sigma=\\hat\\Sigma_2+D_2>0$ for a diagonal $D_2$.\nThen,\\begin{align}\\label{trmc2}\n &(\\hat\\Sigma_2,D_2)=\\arg\\min\\{ {\\operatorname{trace}}(\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq 0,\\;\\mbox{diagonal }D\\}\\\\\n& \\Leftrightarrow~\\exists~ \\Lambda_2 \\geq0 ~:~ \\hat\\Sigma_2 \\Lambda_2=0 \\text{~and~} [\\Lambda_2]_{ii}=1~ \\forall i.\\nonumber\n\\end{align}\n\\end{prop}\n\\end{subequations}\n\n\\noindent Evidently, when the solutions to these two problems differ\nand $D_1\\neq D_2$, then there exists $k\\in\\left\\{1, \\ldots, n\\right\\}$ such that\n\\[\n[D_2]_{kk}<0 \\text{~and~} [D_1]_{kk}=0.\n\\]\nFurther, the essence of Proposition \\ref{prop:MTFAShapiro} is that\na singular $\\hat\\Sigma$ originates from such a minimization problem if and only if there is a correlation matrix in its null space. The matrices $\\Lambda_1$ and $\\Lambda_2$ appear as Lagrange multipliers in the respective problems.\n\n\\newcommand{{\\mathcal A}}{{\\mathcal A}}\n{Factor analysis is closely related to {\\em low-rank matrix completion} as well as to {\\em sparse and low-rank decomposition} problems. Typically, low-rank matrix completion asks for a matrix $X$ which satisfies a linear constraint ${\\mathcal A}(X)=b$ and has low\/minimal rank (${\\mathcal A}(\\cdot)$ denotes a linear map ${\\mathcal A}\\,:\\,{{\\mathbb R}}^{n\\times n}\\rightarrow {{\\mathbb R}}^p$). Thus, factor analysis corresponds to the special case where ${\\mathcal A}(\\cdot)$ maps $X$ onto its off-diagonal entries. In a recent work by Recht {\\em etal.}~\\cite{Recht2010guaranteed}, the nuclear norm of $X$ was considered as a convex relaxation of ${\\operatorname{rank}}(X)$ for such problems and a sufficient condition for exact recovery was provided. However, this sufficient condition amounts to the requirement that the null space of ${\\mathcal A}(\\cdot)$ contains no matrix of low-rank. Therefore, since in factor analysis diagonal matrices are in fact contained in the null space of ${\\mathcal A}(\\cdot)$ and include matrices of low-rank, the condition in \\cite{Recht2010guaranteed} does not apply directly. Other works on low-rank matrix completion (see, e.g., \\cite{Recht2010guaranteed,Candes2009exact}) mainly focus on assessing the probability of exact recovery and on constructing efficient computational algorithms for {\\em large-scale} low-rank completion problems \\cite{Keshavan2010matrix,Keshavan2010noisy}.\nOn the other hand, since diagonal matrices are sparse (most of their entries are zero), the work on matrix decomposition into sparse and low-rank components by Chandrasekaran {\\em etal.} \\cite{Chandrasekaran2011rank} is very pertinent. In this, the $\\ell_1$ and nuclear norms were used as surrogates for sparsity and rank, respectively, and a sufficient condition for exact recovery was provided which captures a certain ``rank-sparsity incoherence''; an analogous but stronger sufficient ``incoherence'' condition which applies to problem\n\\eqref{trmc2} is given in \\cite{Saunderson2012}.}\n\n\n\\subsection{Weighted minimum trace factor analysis}\n\nBoth ${\\operatorname{mr}}(\\Sigma)$ and ${\\operatorname{mr}}_+(\\Sigma)$ in \\eqref{eq:mc} and\n\\eqref{eq:mc2}, respectively, remain invariant under scaling of rows\nand the corresponding columns of $\\Sigma$ by the same coefficients.\nOn the other hand, the minimizers in \\eqref{trmc} and \\eqref{trmc2}\nand their respective ranks are not invariant under scaling. This\nfact motivates weighted-trace minimization,\n\\begin{align}\\label{eq:Dw}\n\\min\\left\\{ {\\operatorname{trace}}(W\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D,~\\hat\\Sigma\\geq 0,~\\mbox{diagonal }D\\geq 0 \\right\\},\n\\end{align}\ngiven $\\Sigma>0$ and a diagonal weight $W>0$.\nAs before the characterization of minimizers relates to a suitable condition for the corresponding Lagrange multipliers:\n\n\\begin{prop}[{\\rm \\cite{Shapiro1982b}}]\\label{prop:WMTFAShapiro}\nLet $\\Sigma=\\hat\\Sigma_0+D_0>0$ for a diagonal matrix $D_0\\geq0$ and consider a diagonal $W>0$. Then,\n\\begin{align}\\label{trmc3}\n &(\\hat\\Sigma_0,D_0)=\\arg\\min\\{ {\\operatorname{trace}}(W\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq\n 0,\\;\\mbox{diagonal }D\\geq 0\\}\\\\\n& \\Leftrightarrow~ \\exists~ \\Lambda_0 \\geq0 ~:~\n\\hat\\Sigma \\Lambda_0=0 \\text{~and~} \\left\\{\n \\begin{array}{ll}\n [\\Lambda_0]_{ii}=[W]_{ii}, & \\text{~if~} [D_0]_{ii}>0, \\\\\n \\left[ \\Lambda_0\\right]_{ii}\\geq [W]_{ii}, &\\text{~if~}[D_0]_{ii}=0.\\nonumber\n \\end{array}\n \\right.\\nonumber\n\\end{align}\n\\end{prop}\n\nA corresponding sufficient and necessary condition for $(\\hat\\Sigma, D)$ to be a minimizer in Shapiro's problem is that there exists a Grammian in the null space of $\\hat\\Sigma$ whose diagonal entries are equal to the diagonal entries of $W$.\n\nMinimum-rank solutions may be recovered as solutions to \\eqref{trmc3} using suitable choices of weight.\nHowever, these choices depend on $\\Sigma$ and are not known in advance --this motivates a selection of certain canonical $\\Sigma$-dependent\nweight as well as iteratively improving the choice of weight. One should note that since $D$ is diagonal, letting $W$ be a not-necessarily\ndiagonal matrix does not change the problem --only the diagonal entries of $W$ determine the minimizer.\n\nWe first consider taking $W=\\Sigma^{-1}$. A rationale for this\nchoice is that the minimal value in \\eqref{eq:Dw} bounds\n${\\operatorname{mr}}_+(\\Sigma)$ from below, since for any decomposition\n$\\Sigma=\\hat\\Sigma+D$,\n\\begin{align}\\nonumber\n{\\operatorname{rank}}(\\hat \\Sigma) =&~ {\\operatorname{trace}} (\\hat\\Sigma^\\sharp \\hat\\Sigma)\\\\\\nonumber\n\\geq&~ {\\operatorname{trace}}((\\hat\\Sigma+D)^{-1} \\hat\\Sigma)\\\\\n=&~ {\\operatorname{trace}}(\\Sigma^{-1} \\hat\\Sigma)\\label{eq:rankjustify}\n\\end{align}\nwhere $^\\sharp$ denotes the Moore-Penrose pseudo inverse. Continuing\nwith this line of analysis\n\\begin{align}\n{\\operatorname{rank}}(\\hat \\Sigma) =&~ {\\operatorname{trace}} (\\hat\\Sigma^\\sharp \\hat\\Sigma)\\nonumber\\\\\n\\geq&~ {\\operatorname{trace}}((\\hat\\Sigma+\\epsilon I)^{-1} \\hat\\Sigma)\\label{eq:ranktrace}\n\\end{align}\nfor any $\\epsilon>0$, suggests the iterative re-weighting process\n\\begin{align}\\label{minimizer_a}\nD_{(k+1)}:=&~\\arg\\min_{D}{\\operatorname{trace}}\\left((\\Sigma-D_{(k)} +\\epsilon I)^{-1}(\\Sigma-D)\\right)\n\\end{align}\nfor $k=1,\\,2,\\,\\ldots$ and $D_{(0)}:=0$.\nIn fact, as pointed out in \\cite{Fazel2003}, \\eqref{minimizer_a} corresponds to minimizing\n$\\log\\det(\\Sigma-D+\\epsilon I)$\nby local linearization.\n\nNext we provide a sufficient condition for $\\hat\\Sigma$ to be such a\nstationary point \\eqref{minimizer_a}, i.e., \nfor $\\hat\\Sigma$ to satisfy\n\\begin{align}\\label{stationary_a}\n\\arg\\min_{D}{\\operatorname{trace}}\\left((\\hat\\Sigma+\\epsilon I)^{-1}(\\hat\\Sigma-D)\\right)=0.\n\\end{align}\nThe notation $\\circ$ used below denotes the\nelement-wise product between vectors or matrices which is also known\nas \\emph{Schur product} \\cite{Horn1990matrix} and, likewise, for\nvectors $a, b \\in {\\mathbb R}^{n\\times 1}$, $a\\circ b\\in {\\mathbb R}^{n\\times 1}$\nwith $[a\\circ b]_i=[a]_i[b]_i$.\n\n\n\\begin{prop}\\label{prop:stationary_a}\nLet $\\hat\\Sigma\\in{\\mathbf S}_{n,+}$ and let the columns of $U$ form a basis of ${\\mathcal R}(\\hat\\Sigma)$. If\n\\begin{align}\\label{eq:stationary_a}\n{\\mathcal R}(U\\circ U) \\subset {\\mathcal R}(\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\circ\\Pi_{{\\mathcal N}(\\hat\\Sigma)} ),\n\\end{align}\nthen $\\hat\\Sigma$ satisfies \\eqref{stationary_a} for all $\\epsilon\\in(0,\\; \\epsilon_1)$ and some $\\epsilon_1>0$.\n\\end{prop}\n\nWe first need the following result which generalizes \\cite[Theorem 3.1]{Shapiro1985}.\n\\begin{lemma}\\label{lemma:trace}\nFor $A\\in {\\mathbb R}^{n\\times p}$ and $B\\in {\\mathbb R}^{n\\times q}$ having columns $a_1, \\ldots, a_p$ and $b_1, \\ldots, b_q$, respectively, we let\n\\begin{align*}\nC&=[a_1\\circ b_1, a_1\\circ b_2, \\ldots,a_2\\circ b_1\\dots a_p \\circ b_q]\\in {\\mathbb R}^{n\\times pq},\\\\\n\\phi &: ~{\\mathbb R}^n \\hspace*{.4cm}\\rightarrow {\\mathbb R}^n \\hspace*{.57cm} d \\mapsto {\\operatorname{diag}}(AA'{\\operatorname{diag}}^*(d)BB'), \\mbox{ and}\\\\\n\\psi &: ~{\\mathbb R}^{p\\times q} \\rightarrow {\\mathbb R}^n \\hspace*{.5cm} \\Delta \\mapsto {\\operatorname{diag}}(A\\Delta B').\n\\end{align*}\nThen ${\\mathcal R}(\\phi)={\\mathcal R}(\\psi)={\\mathcal R}((AA')\\circ (BB'))={\\mathcal R}(C)$.\n\\end{lemma}\n\n\\begin{proof}\nSince ${\\operatorname{diag}}(AA'{\\operatorname{diag}}^*(d)BB')=((AA')\\circ (BB'))d$, it follows that\n\\[{\\mathcal R}(\\phi)={\\mathcal R}((AA')\\circ (BB').\\]\nMoreover,\n${\\operatorname{diag}}(A\\Delta B')= \\sum_{i=1}^p\\sum_{j=1}^q a_i\\circ b_j [\\Delta]_{ij}$, and then\n${\\mathcal R}(\\psi)={\\mathcal R}(C)$.\nWe only need to show that ${\\mathcal R}(C)={\\mathcal R}((AA')\\circ (BB'))$. This follows from\n\\begin{align*}\n(AA')\\circ (BB') =&~\\sum_{i=1}^p\\sum_{j=1}^q (a_ia_i')\\circ (b_jb_j')\\\\\n =&~\\sum_{i=1}^p\\sum_{j=1}^q (a_i\\circ b_j) (a_i\\circ b_j)'\n =CC'.\n\\end{align*}\nThus ${\\mathcal R}(C)={\\mathcal R}((AA')\\circ (BB'))$.\n\\end{proof}\n\n\n\\begin{proof} {\\em [Proof of Proposition \\ref{prop:stationary_a}:]}\nAssume that $\\hat\\Sigma$ satisfies \\eqref{stationary_a}.\nIf ${\\operatorname{rank}}(\\hat\\Sigma)=r$, let $\\hat\\Sigma=USU'$ be the eigendecomposition of $\\hat\\Sigma$ with $S={\\operatorname{diag}}^*(s)$ with $s\\in {\\mathbb R}^r$. Let the columns of $V$ be an orthogonal basis of the null space of $\\hat\\Sigma$, i.e., $\\Pi_{{\\mathcal N}(\\hat\\Sigma)}=VV'$.\nThen\n\\begin{align*}\n(\\hat\\Sigma+\\epsilon I)^{-1}=(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)}+\\epsilon \\Pi_{{\\mathcal N}(\\hat\\Sigma)})^{-1} =(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\frac{1}{\\epsilon} \\Pi_{{\\mathcal N}(\\hat\\Sigma)},\n\\end{align*}\nand\n\\begin{align*}\n\\arg\\min_{D:\\hat\\Sigma\\geq D} {\\operatorname{trace}}\\left((\\hat\\Sigma+\\epsilon I)^{-1}(\\hat\\Sigma-D)\\right)& =\\\\\n&\\hspace*{-1.5cm}\\arg\\min_{D:\\hat\\Sigma\\geq D} {\\operatorname{trace}}\\left(\\left(\\epsilon( \\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\right)(\\hat\\Sigma-D)\\right).\n\\end{align*}\nFrom Proposition \\ref{prop:WMTFAShapiro}, \\eqref{stationary_a} holds if there is $M\\in {\\mathbf S}_{r,+}$ such that\n\\begin{align}\\label{stationaryaDiag}\n{\\operatorname{diag}}(VMV')={\\operatorname{diag}}\\left( \\epsilon( \\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\right).\n\\end{align}\nObviously, if $\\epsilon=0$\n $M=I$ satisfies the above equation. We consider the matrix $M$ of the form $M=I+\\Delta$. For \\eqref{stationaryaDiag} holds, we need ${\\operatorname{diag}}((\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}})^\\sharp)$ to be in the range of $\\psi$ for\n\\[\n\\psi: {\\mathbf S}_n \\rightarrow {\\mathbb R}^n \\hspace*{.57cm} \\Delta \\mapsto {\\operatorname{diag}}(V\\Delta V').\n\\]\nFrom Lemma \\ref{lemma:trace} that ${\\mathcal R}(\\psi)={\\mathcal R}(\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\circ\\Pi_{{\\mathcal N}(\\hat\\Sigma)})$. On the other hand, since\n\\[\n\\epsilon(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp=U{\\operatorname{diag}}\\left(\\left[\\frac{\\epsilon}{[s]_1+\\epsilon}, \\ldots, \\frac{\\epsilon}{[s]_r+\\epsilon} \\right]\\right)U',\n\\]\nthen ${\\operatorname{diag}}(\\epsilon(\\hat\\Sigma+\\epsilon\n\\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp)\\in {\\mathcal R}(U\\circ U)$. So if\n\\eqref{eq:stationary_a} holds, there is always a $\\Delta$ such that\n$M=I+\\Delta$ satisfies \\eqref{stationaryaDiag}. Morover, it is also\nrequired that $I+\\Delta\\geq0$. Since the map from $\\epsilon$ to\n$\\Delta$ is continuous, for small enough $\\epsilon$, i.e. in a\ninterval $(0, \\epsilon_1)$ the condition $I+\\Delta$ can always be\nsatisfied.\n\\end{proof}\n\nWe note that \\eqref{eq:stationary_a} is a sufficient condition for $\\hat\\Sigma$ to be a stationary point of \\eqref{stationary_a} in both Frisch's and Shapiro's settings.\n\n\n\\section{Certificates of minimum rank}\\label{sec:CertifMinRank}\n\nWe are interested in obtaining bounds on the minimal rank for the\nFrisch problem so as to ensure optimality when candidate solutions\nare obtained by the earlier optimization approach in \\eqref{minimizer_a}.\n\nThe following two bounds were proposed in \\cite{Woodgate1}, and\nfollow from Theorem~\\ref{thm:Reiersol}. However, both of these\nbounds require exhaustive search which may be prohibitively\nexpensive when $n$ is large.\n\\begin{subequations}\n\\begin{cor}\\label{cor1}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ and $\\Sigma>0.$ If there is an $s_1\\times s_1$\nprinciple minor of $\\Sigma$ whose inverse is positive, then\n \\begin{align}\n{\\operatorname{mr}}_+(\\Sigma)&\\geq s_1-1.\n \\end{align}\n If there is an $s_2\\times s_2$ principle\nminor of $\\Sigma^{-1}$ which is element-wise positive, then\n \\begin{align}\n{\\operatorname{mr}}_+(\\Sigma)&\\geq s_2-1.\n \\end{align}\n\\end{cor}\n\nNext we discuss three other bounds that are computationally\nmore tractable --the first two were proposed by Guttman\n\\cite{Guttman1954}.\nGuttman's bounds are based on a conservative assessment for the admissible range of each of the diagonal entries of $D=\\Sigma-\\hat\\Sigma$.\n\n\\begin{prop}\\label{prop:Guttman}\n Let $\\Sigma\\in {\\mathbf S}_{n,+}$ and let\n \\begin{align*}\n D_1&:={\\operatorname{diag}}^*({\\operatorname{diag}}(\\Sigma))\\\\\n D_2&:=\\left({\\operatorname{diag}}^*({\\operatorname{diag}}(\\Sigma^{-1}))\\right)^{-1}.\n \\end{align*}\n Then the following hold,\n\\begin{align}\n&{\\operatorname{mr}}_+(\\Sigma)\\geq n_+(\\Sigma-D_1) \\label{Guttman:bound1}\\\\\n&{\\operatorname{mr}}_+(\\Sigma)\\geq n_+(\\Sigma-D_2). \\label{Guttman:bound2}\\\\\n\\nonumber\n\\end{align}\nFurther, $n_+(\\Sigma-D_1)\\leq n_+(\\Sigma-D_2)$.\n\\end{prop}\n\n\\begin{proof} The proof follows from the fact that $\\Sigma\\geq D$ implies $D\\leq D_2\\leq D_1$. See \\cite{Guttman1954} for details.\n\\end{proof}\n\nIt is also easy to see that ${\\operatorname{mr}}(\\Sigma)\\geq n_+(\\Sigma-D_1)$ which\nprovides a lower bound for the minimum rank in Shapiro's problem.\nNext we return to a bound, which we noted earlier in \\eqref{eq:rankjustify}.\n\n\\begin{prop}\\label{tracebound}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$. Then the following holds:\n\\begin{align}\\label{eq:lowerbound3}\n{\\operatorname{mr}}_+(\\Sigma)\\geq \\min_{\\Sigma\\geq D\\geq 0}{\\operatorname{trace}}(\\Sigma^{-1}(\\Sigma-D)).\n\\end{align}\n\\end{prop}\n\n\\begin{proof} The statement follows readily from \\eqref{eq:rankjustify}.\n\\end{proof}\n\nEvidently an analogous statement holds for ${\\operatorname{mr}}(\\Sigma)$.\nWe note that \\eqref{Guttman:bound1} and \\eqref{Guttman:bound2} remain invariant\nunder scaling of rows and corresponding columns, whereas \\eqref{eq:lowerbound3} does not, hence these two cannot be compared directly.\n\\end{subequations}\n\n\\section{Correspondence between decompositions}\\label{correspondence}\n\nWe now return to the decomposition of the data matrix $X=\\hat\nX+\\tilde X$ as in \\eqref{eq:decompose} and its relation to the\ncorresponding sample covariances. The decomposition of $X$ into\n``noise-free'' and ``noisy'' components implies a corresponding\ndecomposition for the sample covariance, but in the converse\ndirection, a decomposition $ \\Sigma=\\hat\\Sigma+\\tilde\\Sigma $ leads\nto a family of compatible decompositions for $X$, which corresponds\nto the boundary of a matrix-ball. This is discussed next.\n\n\\begin{prop}\\label{prop:decomposition} Let $X\\in{\\mathbb R}^{n\\times T}$, and $\\Sigma:=XX^\\prime$. If\n\\begin{equation}\\label{eq:decompose2}\n\\Sigma=\\hat \\Sigma+\\tilde \\Sigma\n\\end{equation}\nwith $\\hat \\Sigma$, $\\tilde \\Sigma$ symmetric and non-negative definite, there exists a decomposition\n\\begin{subequations}\\label{conditions}\n\\begin{equation}\\label{eq:Xdecompose}\nX=\\hat X+\\tilde X\n\\end{equation}\nfor which\n\\begin{eqnarray}\n\\label{cond2}\n&&\\hat X \\tilde X^\\prime = 0,\\\\\\label{cond3}\n&&\\hat \\Sigma = \\hat X \\hat X^\\prime,\\\\\\label{cond4}\n&&\\tilde \\Sigma = \\tilde X\\tilde X^\\prime.\n\\end{eqnarray}\n\\end{subequations}\nFurther,\nall pairs $(\\hat X,\\,\\tilde X)$ that satisfy (\\ref{eq:Xdecompose}-\\ref{cond4})\nare of the form\n\\begin{equation}\\label{parametrization}\n\\hat X=\\hat\\Sigma \\Sigma^{-1} X+R^{1\/2}V,\\;\n\\tilde X=\\tilde\\Sigma \\Sigma^{-1} X-R^{1\/2}V,\n\\end{equation}\nwith\n\\begin{subequations}\\label{Rs}\n\\begin{eqnarray}\\label{R}\nR&:=&\\hat \\Sigma - \\hat \\Sigma \\Sigma^{-1} \\hat \\Sigma\\\\\n&=&\\tilde \\Sigma - \\tilde \\Sigma \\Sigma^{-1} \\tilde \\Sigma \\label{R2}\\\\\\nonumber\n&=&\\hat \\Sigma \\Sigma^{-1}\\tilde \\Sigma\\\\\\nonumber\n&=&\\tilde \\Sigma \\Sigma^{-1}\\hat \\Sigma,\\nonumber\n\\end{eqnarray}\n\\end{subequations}\nand $V\\in{\\mathbb R}^{n\\times T}$ such that $VV'=I$, $XV'=0$.\n\\end{prop}\n\n\n\\begin{proof} The proof relies on a standard lemma (\\cite[Theorem 2]{douglas}) which states that if\n $A\\in{\\mathbb R}^{n\\times T}$, $B\\in{\\mathbb R}^{n\\times m}$ with $m\\leq T$ such that\n$A A^\\prime = B B^\\prime,$\nthen $A=BU$ for some $U\\in{\\mathbb R}^{m\\times T}$ with $U U^\\prime =I$.\nThus, we let $A:=X$,\n\\[\nS:=\\left[\\begin{matrix}\\hat \\Sigma & 0\\\\0&\\tilde \\Sigma\\end{matrix}\\right],\n\\]\nand $B:=\\left[\\begin{matrix}I&I \\end{matrix}\\right] S^{1\/2}$,\nwhere $S^{1\/2}$ is the matrix-square root of $S$.\nIt follows that there exists a matrix $U$ as above for which $A=BU$, and therefore we can take\n\\[\n\\left[\\begin{matrix}\\hat X \\\\ \\tilde X\\end{matrix}\\right]:=S^{1\/2}U.\n\\]\nThis establishes the existence of the decomposition\n\\eqref{eq:Xdecompose}.\n\nIn order to parameterize all such pairs $(\\hat X,\\,\\tilde X)$, let\n$U_o$ be an orthogonal (square) matrix such that\n\\[XU_o=[\\Sigma^{1\/2} \\; 0].\n\\]\nThen $\\hat X U_o$ and $\\tilde X U_o$ must be of the form\n\\begin{equation}\\label{UXs}\n\\hat X U_o=: \\left[\\begin{matrix}\\hat X_1&\\Delta \\end{matrix}\\right],\\;\n\\tilde X U_o=: \\left[\\begin{matrix}\\tilde X_1& -\\Delta \\end{matrix}\\right],\n\\end{equation}\nwith $\\hat X_1$, $\\tilde X_1$ square matrices. Since\n\\[\\left[\\begin{matrix}\\hat X\\\\\\tilde X \\end{matrix}\\right]\n\\left[\\begin{matrix}\\hat X^\\prime &\\tilde X^\\prime \\end{matrix}\\right]\n=\\left[\\begin{matrix}\\hat \\Sigma& 0\\\\0&\\tilde \\Sigma \\end{matrix}\\right],\n\\]\nthen\n\\begin{subequations}\n\\begin{eqnarray}\\label{first}\n&&\\hat X_1\\hat X_1^\\prime+\\Delta\\Delta^\\prime=\\hat \\Sigma\\\\\\label{second}\n&&\\hat X_1\\tilde X_1^\\prime-\\Delta\\Delta^\\prime=0\\\\\\label{third}\n&&\\tilde X_1\\tilde X_1^\\prime+\\Delta\\Delta^\\prime=\\tilde \\Sigma.\n\\end{eqnarray}\n\\end{subequations}\nSubstituting $\\hat X_1\\tilde X_1^\\prime$ for $\\Delta\\Delta^\\prime$ into (\\ref{first}) and\nusing the fact that $\\tilde X_1=X_1-\\hat X_1$ with $X_1=\\Sigma^{1\/2}$ we obtain that\n\\begin{eqnarray*}\n&&\\hat X_1=\\hat \\Sigma\\Sigma^{-1\/2}.\n\\end{eqnarray*}\nSimilarly, using (\\ref{third}) instead, we obtain that\n\\begin{eqnarray*}\n&&\\tilde X_1=\\tilde \\Sigma\\Sigma^{-1\/2}.\n\\end{eqnarray*}\nSubstituting into (\\ref{second}), (\\ref{first}) and (\\ref{third}) we obtain the following three relations\n\\begin{eqnarray*}\n\\Delta\\Delta^\\prime &=& \\hat \\Sigma \\Sigma^{-1}\\tilde \\Sigma\\\\\n&=&\\hat \\Sigma - \\hat \\Sigma \\Sigma^{-1} \\hat \\Sigma\\\\\n&=&\\tilde \\Sigma - \\tilde \\Sigma \\Sigma^{-1} \\tilde \\Sigma.\n\\end{eqnarray*}\nSince $\\Delta\\Delta^\\prime$ and the $\\Sigma$'s are all symmetric,\n\\begin{eqnarray*}\n\\Delta\\Delta^\\prime&=&\\tilde \\Sigma \\Sigma^{-1}\\hat \\Sigma\n\\end{eqnarray*}\nas well. Thus, $\\Delta=R^{1\/2}V_1$ with $V_1V_1^\\prime=I$. The proof\nis completed by substituting the expressions for $\\hat X_1$ and\n$\\Delta$ into \\eqref{UXs}.\n\\end{proof}\n\nInterestingly,\n\\[\n{\\operatorname{rank}}(R)+{\\operatorname{rank}}(\\Sigma)={\\operatorname{rank}} \\left(\\left[\n \\begin{array}{cc}\n \\hat\\Sigma & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\Sigma \\\\\n \\end{array}\n \\right] \\right)={\\operatorname{rank}} \\left(\\left[\n \\begin{array}{cc}\n \\hat\\Sigma & 0 \\\\\n 0 & \\tilde\\Sigma \\\\\n \\end{array}\n \\right] \\right)={\\operatorname{rank}}(\\hat\\Sigma)+{\\operatorname{rank}}(\\tilde\\Sigma),\n\\]\nand hence, the rank of the ``uncertainty radius'' $R$ of the corresponding $\\hat X$ and $\\tilde X$-matrix spheres is\n\\[{\\operatorname{rank}}(R)= {\\operatorname{rank}}(\\hat\\Sigma)+{\\operatorname{rank}}(\\tilde\\Sigma)-{\\operatorname{rank}}(\\Sigma).\n\\]\nIn cases where identifying $\\hat X$ from the data matrix $X$,\ndifferent criteria may be used to quantify uncertainty. One such is\nthe rank of $R$ while another is its trace, which is the variance of\nestimation error in determining $\\hat X$. This topic is considered\nnext and its relation to the Frisch decomposition highlighted.\n\n\\section{Uncertainty and worst-case estimation}\\label{sec:min-max}\nThe basic premise of the decomposition (\\ref{eq:decompose2}) is\nthat, in principle, no probabilistic description of the data is\nneeded. Thus, under the assumptions of\nProposition~\\ref{prop:decomposition}, $R$ represents a deterministic\nradius of uncertainty in interpreting the data. On the other hand,\nwhen data and noise are probabilistic in nature and represent\nsamples of jointly Gaussian random vectors ${\\mathbf x},\\;{\\hat{\\mathbf x} },\\; {\\tilde{\\mathbf x}}$ as\nin (\\ref{eq:xa} - \\ref{eq:xc}), the conditional expectation of\n${\\hat{\\mathbf x} }$ given ${\\mathbf x}$ is $E\\{{\\hat{\\mathbf x} } |{\\mathbf x}\\}=\\hat\\Sigma\\Sigma^{-1} {\\mathbf x}$,\nwhile the variance of the error\n\\begin{eqnarray*}\nE\\{({\\hat{\\mathbf x} }-\\hat\\Sigma \\Sigma^{-1}{\\mathbf x})({\\hat{\\mathbf x} }-\\hat\\Sigma \\Sigma^{-1}{\\mathbf x})^\\prime\\}&=&\\hat\\Sigma - \\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\\\\\n&=&R\n\\end{eqnarray*}\nis the radius of the deterministic uncertainty set. Either way, it is of interest to assess how this radius depends on the decomposition of $\\Sigma$.\n\n\\subsection{Uniformly optimal decomposition}\n\nSince the decomposition of $\\Sigma$ in the Frisch problem is not\nunique, it is natural to seek a uniformly optimal choice of the\nestimate $K{\\mathbf x}$ for ${\\hat{\\mathbf x} }$ over all admissible decompositions. To\nthis end, we denote the mean-squared-error loss function\n\\begin{eqnarray}\\label{eq:LossFunction}\nL(K, \\hat \\Sigma, \\tilde\\Sigma)&:=&{\\operatorname{trace}}\\left({\\mathcal E}\\left( ({\\hat{\\mathbf x} }-K{\\mathbf x})({\\hat{\\mathbf x} }-K{\\mathbf x})^\\prime\\right)\\right)\\nonumber\\\\\n&\\;=&{\\operatorname{trace}}\\left(\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K(\\hat\\Sigma+\\tilde\\Sigma) K' \\right),\\label{eq:loss}\n\\end{eqnarray}\nand define\n\\begin{align*}\n{\\mathcal S}(\\Sigma):= \\{(\\hat\\Sigma, \\tilde\\Sigma) : &~\\Sigma=\\hat\\Sigma+\\tilde\\Sigma,\\; \\hat\\Sigma,\\; \\tilde\\Sigma\\geq0 \\text{~and~} \\tilde\\Sigma \\text{~is diagonal} \\}\n\\end{align*}\nas the set of all admissible pairs. Thus, a uniformly-optimal\ndecomposition of $X$ into signal plus noise relates to the following\nmin-max problem:\n\\begin{align}\\label{prob:minmax}\n\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma, \\tilde\\Sigma).\n\\end{align}\nThe minimizer of \\eqref{prob:minmax} is the uniformly optimal\nestimator gain $K$. Analogous min-max problems, over\ndifferent uncertainty sets, have been studied in the literature\n\\cite{Eldar2004competitive}. In our setting\n\\begin{subequations}\\label{eq:concave}\n\\begin{eqnarray}\n\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma,\\tilde\\Sigma)&\\geq&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma,\\tilde\\Sigma)\\label{minmaxmaxmin}\\\\\n&=&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left(\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\\right)\\label{eq:concave1}\\\\\n&=&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left(\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma\\right).\\label{eq:concave2}\n\\end{eqnarray}\n\\end{subequations}\nThe functions to maximize in \\eqref{eq:concave1} and\n\\eqref{eq:concave2} are both strictly concave in $\\hat\\Sigma$ and\n$\\tilde\\Sigma$. Therefore the maximizer is unique. Thus, we denote\n\\begin{equation}\\label{optsolution}\n(K_{\\rm opt}, \\hat\\Sigma_{\\rm opt}, \\tilde\\Sigma_{\\rm opt}) :=\\arg \\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma,\\tilde\\Sigma),\n\\end{equation}\nwhere, clearly, $K_{\\rm opt}=\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}$.\n\nIn general, the decomposition suggested by the uniformly optimal\nestimation problem does not lead to a singular signal covariance\n$\\hat\\Sigma$. The condition for when that happens is given next.\nInterestingly, this is expressed in terms of half the candidate\nnoise covariance utilized in obtaining one of the Guttman bounds\n(Proposition \\ref{prop:Guttman}).\n\n\\begin{prop}\\label{prop:maxmin}\nLet $\\Sigma>0$, and let\n\\begin{equation}\\label{eq:D0}\nD_0:=\\frac12 {\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1}\n\\end{equation}\n(which is equal to $\\frac12 D_2$ defined in Proposition \\ref{prop:Guttman}).\nIf $\\Sigma-D_0\\geq0$, then\n\\begin{subequations}\\label{Solution}\n\\begin{equation}\\label{InteriorSolution}\n\\tilde\\Sigma_{\\rm opt}=D_0 \\text{~and~} \\hat\\Sigma_{\\rm opt}=\\Sigma-D_0.\n\\end{equation}\nOtherwise,\n\\begin{equation}\\label{BoundarySolution}\n\\tilde\\Sigma_{\\rm opt}\\leq D_0 \\text{~and~} \\hat\\Sigma_{\\rm opt} \\text{~is singular}.\n\\end{equation}\n\\end{subequations}\n\\end{prop}\n\n\\begin{proof}\nFrom \\eqref{eq:concave2},\n\\begin{eqnarray}\nL(K_{\\rm opt}, \\hat \\Sigma_{\\rm opt}, \\tilde \\Sigma_{\\rm opt})&=&\\max \\left\\{\\tilde \\Sigma-\\tilde \\Sigma\\Sigma^{-1}\\tilde \\Sigma ~\\mid~ \\Sigma\\geq\\tilde\\Sigma\\geq0, \\tilde\\Sigma \\text{~is diagonal} \\right\\}\\nonumber\\\\\n&\\leq& \\max \\left\\{\\tilde \\Sigma-\\tilde \\Sigma\\Sigma^{-1}\\tilde \\Sigma ~\\mid~\\tilde\\Sigma \\text{~is diagonal} \\right\\}\\label{relaxedD}\\\\\n&=&\\frac12 {\\operatorname{trace}}(D_0)\\nonumber\n\\end{eqnarray}\nwith the maximum attained for $\\tilde\\Sigma=D_0$. Then\n\\eqref{InteriorSolution} follows. In order to prove\n\\eqref{BoundarySolution}, consider the Lagrangian corresponding to\n\\eqref{eq:concave2}\n\\[\n{\\mathcal L}(\\tilde\\Sigma,\\Lambda_0, \\Lambda_1) ={\\operatorname{trace}}(\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma+\\Lambda_0(\\Sigma-\\tilde\\Sigma)+\\Lambda_1\\tilde\\Sigma)\n\\]\nwhere $\\Lambda_0,\\;\\Lambda_1$ are Lagrange multipliers.\nThe optimal values satisfy\n\\begin{subequations}\n\\begin{eqnarray}\n&&[I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm opt}-\\Lambda_{0}+\\Lambda_{1}]_{kk}=0, \\;\\forall\\; k=1,\\ldots, n,\\label{condition1}\\\\\n&& \\Lambda_{0}\\hat\\Sigma_{\\rm opt}=0,\\; \\Lambda_{0}\\geq0,\\label{condition2}\\\\\n&& \\Lambda_{1}\\tilde\\Sigma_{\\rm opt}=0,\\; \\Lambda_{1}\\geq0 \\text{~and is diagonal}.\\label{condition3}\n\\end{eqnarray}\n\\end{subequations}\nIf $\\Sigma- D_0\\not\\geq0$ we show that $\\hat\\Sigma_{\\rm opt}$ is\nsingular. Assume the contrary, i.e., that $\\hat\\Sigma_{\\rm opt}>0$.\nFrom \\eqref{condition2}, we see that $\\Lambda_{0}=0$, while from\n\\eqref{condition1}, $ [I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm\nopt}]_{kk}\\leq 0. $ This gives that\n\\[\n[\\tilde\\Sigma_{\\rm opt}]_{kk}\\geq \\frac{1}{2[\\Sigma^{-1}]_{kk}}= [D_0]_{kk},\n\\]\nfor all $k=1, \\ldots, n$, which contradicts the fact that $\\Sigma-D_0\\not\\geq0$. Therefore $\\hat\\Sigma_{\\rm opt}$ is singular.\nWe now assume that $\\tilde\\Sigma\\not \\leq D_0$. Then there exists $k$ such that $[\\tilde\\Sigma_{\\rm opt}]_{kk}> [D_0]_{kk}$.\nFrom \\eqref{condition3} and \\eqref{condition1}, we have that\n \\[\n [\\Lambda_{1}]_{kk}=0 \\text{~and~} [I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm opt}]_{kk}\\geq0\n \\]\nwhich contradicts the assumption that $[\\tilde\\Sigma_{\\rm opt}]_{kk}>\n[D_0]_{kk}$. Therefore $\\tilde\\Sigma_{\\rm opt}\\leq D_0$ and\n\\eqref{BoundarySolution} has been established.\n\\end{proof}\n\nWe remark that while\n\\begin{eqnarray*}\n{\\mathcal E}\\left( ({\\hat{\\mathbf x} }-K{\\mathbf x})({\\hat{\\mathbf x} }-K{\\mathbf x})^\\prime\\right)&=&\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K\\Sigma K'\\\\\n&=&(\\hat\\Sigma\\Sigma^{-\\frac12}-K\\Sigma^{\\frac12})(\\hat\\Sigma\\Sigma^{-\\frac12}-K\\Sigma^{\\frac12})^\\prime+\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\n\\end{eqnarray*}\nis matrix-convex in $K$ and a unique minimum for\n$K=\\hat\\Sigma\\Sigma^{-1}$, the error covariance\n$\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma $ may not have a unique\nmaximum in the positive semi-definite sense. To see this, consider\n$\\Sigma=\\left[\n \\begin{array}{cc}\n 2 & 1 \\\\\n 1 & 2 \\\\\n \\end{array}\n \\right]\n$. In this case $D_0=\\frac{3}{4}I$, $\\hat\\Sigma_{\\rm opt}=\\left[\n \\begin{array}{cc}\n 5\/4 & 1 \\\\\n 1 & 5\/4 \\\\\n \\end{array}\n \\right]$, and\n\\begin{equation}\\label{eq:Ropt}\n\\hat\\Sigma_{\\rm opt}-\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}=\\left[\n \\begin{array}{cc}\n 3\/8 & 3\/16 \\\\\n 3\/16 & 3\/8 \\\\\n \\end{array}\n \\right].\n\\end{equation}\nOn the other hand, for $\\hat\\Sigma=\\left[\n \\begin{array}{cc}\n 3\/2 & 1 \\\\\n 1 & 3\/2 \\\\\n \\end{array}\n \\right]$, then\n\\[\n\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma=\\left[\n \\begin{array}{cc}\n 1\/3 & 1\/12 \\\\\n 1\/12 & 1\/3 \\\\\n \\end{array}\n \\right]\n\\]\nwhich is neither larger nor smaller than \\eqref{eq:Ropt} in the\nsense of semi-definiteness. This is a key reason for considering\nscalar loss functions of the error covariance as in\n\\eqref{eq:loss}.\n\nNext we note that there is no gap between the min-max and max-min\nvalues in the two sides of \\eqref{minmaxmaxmin}.\n\\begin{prop}\\label{prop:minmax}\nFor $\\Sigma\\in{\\mathbf S}_{n,+}$, then\n\\begin{equation}\\label{eq:equal}\n\\min_{K}\\max_{(\\hat\\Sigma, \\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma, \\tilde\\Sigma)=\\max_{(\\hat\\Sigma, \\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma, \\tilde\\Sigma).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe observe that for a fixed $K$, the function $L(K, \\hat \\Sigma,\n\\tilde\\Sigma)$ is a linear function of $(\\hat\\Sigma, \\tilde\\Sigma)$.\nFor fixed $(\\hat\\Sigma, \\tilde\\Sigma)$, the function is a convex\nfunction of $K$. Under this conditions it is standard that\n\\eqref{eq:equal} holds, see e.g. \\cite[page 281]{Boyd2004convex}.\n\\end{proof}\n\n\nWe remark that when $D_0=\\frac12\n{\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1}$ is admissible as noise\ncovariance, i.e., $\\Sigma- D_0\\geq0$, the optimal signal covariance\nis $\\hat\\Sigma_{\\rm opt}=\\Sigma-D_0$, and the gain matrix $K_{\\rm\nopt}=\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}=I-D_0\\Sigma^{-1}$ has all\ndiagonal entries equal to $\\frac{1}{2}$. Thus, with $K_{\\rm opt}$ in\n\\eqref{eq:LossFunction} the mean-square-error loss is independent of\n$\\hat\\Sigma$ and equal to ${\\operatorname{trace}}\\left(K_{\\rm opt}\\Sigma K_{\\rm\nopt}^\\prime\\right)$ for any admissible decomposition of $\\Sigma$.\n\nWe also remark that the key condition (Proposition \\ref{prop:maxmin})\n \\begin{align*}\\label{InvariantCondition}\n& \\Sigma\\geq\\frac12 {\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1}) \\right)^{-1}\\\\\n&\\Leftrightarrow 2{\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1}) \\right)\\geq \\Sigma^{-1}\n \\end{align*}\ncan be equivalently written as $\\Sigma^{-1}\\circ (2I-{\\bf 1}{\\bf\n1}')\\geq0$, and interestingly, amounts to the positive\nsemi-definitess of a matrix formed by changing the signs of all\noff-diagonal entries of $\\Sigma^{-1}$. The set of all such matrices, $\\left\\{S\n\\mid S\\geq 0,~ S\\circ (2I-{\\bf 1}{\\bf 1}')\\geq0 \\right\\}$, is\nconvex, invariant under scaling rows and corresponding columns, and\ncontains the set of diagonally dominant matrices $\\{S \\mid S\\geq 0,~\n[S]_{ii}\\geq \\sum_{j\\neq i} |[S]_{ij}| \\text{~for all ~} i\\}$.\n\nWe conclude this section by noting that ${\\operatorname{trace}}(R_{\\rm opt})$,\n with\n \\[\nR_{\\rm opt}:=\\hat\\Sigma_{\\rm opt}-\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt},\n\\]\nquantifies the distance between admissible decompositions of $\\Sigma$. This is stated next.\n\n\\begin{prop}\\label{lemma:radius}\nFor $\\Sigma>0$ and any pair $(\\hat\\Sigma, \\tilde\\Sigma)\\in {\\mathcal S}(\\Sigma)$,\n\\[\n{\\operatorname{trace}}\\left( (\\hat\\Sigma-\\hat\\Sigma_{\\rm opt})\\Sigma^{-1}(\\hat\\Sigma-\\hat\\Sigma_{\\rm opt})' \\right)\\leq {\\operatorname{trace}}(R_{\\rm opt}).\n\\]\n\\end{prop}\n\\begin{proof}\nClearly\n$0\\leq{\\operatorname{trace}}(\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma)$,\nwhile from Proposition \\ref{prop:minmax},\n\\begin{eqnarray}\nL(K_{\\rm opt}, \\hat\\Sigma, \\tilde\\Sigma)\n&=& {\\operatorname{trace}}(\\hat\\Sigma-2\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma+\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}')\\label{eqB}\\\\\n&\\leq& {\\operatorname{trace}}(R_{\\rm opt}).\\nonumber\n\\end{eqnarray}\nThus, ${\\operatorname{trace}}(\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma-2\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma+\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}')\\leq {\\operatorname{trace}}(R_{\\rm opt})$.\n\\end{proof}\n\n\\subsection{Uniformly optimal estimation and trace regularization}\\label{sec:regularized}\nA decomposition of $\\Sigma$ in accordance with the min-max estimation problem of the previous section often produces an invertible signal covariance $\\hat\\Sigma$. On the other hand, it is often the case and it is the premise of factor analysis, that $\\hat\\Sigma$ is singular of low rank and, thereby, allows identifying linear relations in the data. In this section we consider combining the mean-square-error loss function with regularization term promoting a low rank for the signal covariance $\\hat\\Sigma$ \\cite{Fazel2001}. More specifically, we consider\n\\begin{equation}\\label{prob:minmaxRank}\nJ=\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} \\left(L(K,\n\\hat \\Sigma, \\tilde\\Sigma)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\right),\n\\end{equation}\nfor $\\lambda\\geq0$, and properties of its solutions.\n\nAs noted in Proposition \\ref{prop:minmax} (see \\cite[page 281]{Boyd2004convex}), here too there is no gap between the min-max and the max-min, which becomes\n\\begin{subequations}\n\\begin{align}\n&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma, \\tilde\\Sigma)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\nonumber\\\\\n&= \\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} {\\operatorname{trace}}\\left( (1-\\lambda)\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K(\\hat\\Sigma+\\tilde\\Sigma)K' \\right)\\nonumber\\\\\n&=\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left( (1-\\lambda)\\hat\\Sigma-\\hat\\Sigma(\\hat\\Sigma+\\tilde\\Sigma)^{-1}\\hat\\Sigma \\right) \\label{eq:Kcanceled}\\\\\n&=\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left( -\\lambda\\Sigma+ (1+\\lambda)\\tilde\\Sigma-\\tilde\\Sigma(\\hat\\Sigma+\\tilde\\Sigma)^{-1}\\tilde\\Sigma \\right). \\label{eq:Sigtilde}\n\\end{align}\n\\end{subequations}\nSince \\eqref{eq:Kcanceled}\nand \\eqref{eq:Sigtilde} are strictly concave functions of\n$\\hat\\Sigma$ and $\\tilde\\Sigma$, respectively, there is a unique\nset of optimal values $(K_{\\lambda, \\rm opt}, \\hat\\Sigma_{\\lambda,\\rm opt}, \\tilde\\Sigma_{\\lambda,\\rm opt})$.\n\n\\begin{prop}\nLet $\\Sigma>0$, $D_0=\\frac12 \\left({\\operatorname{diag}}^*{\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1},$\n$\\lambda_{\\rm min}$ be the smallest eigenvalue of $D_0^{-\\frac12}\\Sigma D_0^{-\\frac12}$,\nand $(K_{\\lambda, \\rm opt}, \\hat\\Sigma_{\\lambda,\\rm opt}, \\tilde\\Sigma_{\\lambda,\\rm opt})$ as above, for $\\lambda\\geq0$.\nFor any\n$\\lambda\\geq\\lambda_{\\rm min}-1$,\n$\\hat\\Sigma_{\\lambda,{\\rm opt}}$ is singular.\n\\end{prop}\n\n\\begin{proof}\nThe trace of\n$( -\\lambda\\Sigma+ (1+\\lambda)\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma )$ is maximal for the diagonal choice $\\tilde \\Sigma = (1+\\lambda)D_0$.\nFor any $\\lambda \\geq \\lambda_{\\rm min}-1$, $\\Sigma-(1+\\lambda) D_0$ fails to be positive semidefinite. Thus, the constraint $\\Sigma-\\tilde\\Sigma\\geq 0$ in \\eqref{eq:Sigtilde} is active and $\\hat\\Sigma_{\\lambda, {\\rm opt}}$ is singular.\n\\end{proof}\n\nNote that $\\Sigma-2D_0\\not\\geq 0$ (unless $\\Sigma$ is diagonal), and therefore $\\lambda_{\\rm min}<2$. Hence, for\n$\\lambda\\geq1$, $\\hat\\Sigma_{\\lambda, {\\rm opt}}$ is singular.\nWhen $\\lambda\\to 0$ we recover the solution in \\eqref{optsolution}, whereas for $\\lambda\\to\\infty$ we recover the solution in Proposition\n\\ref{prop:mintrace}.\n\n\n\\section{Accounting for statistical errors}\\label{statisticalerrors}\n\nFrom an applications standpoint $\\Sigma$ represents an empirical\ncovariance, estimated on the basis of a finite observation record in\n$X$. Hence \\eqref{eq:diagonal} and \\eqref{eq:orthogonality} are only\napproximately valid, as already suggested in\nSection~\\ref{sec:datastrcuture}. Thus, in order to account for\nsampling errors we can introduce a penalty for the size of\n $C:=\\hat X\\tilde X^\\prime$, conditioned so that\n \\[\n \\Sigma=\\hat\\Sigma + \\tilde\\Sigma +C +C',\n \\]\nand a penalty for the distance of $\\tilde \\Sigma$ from the set $\\{D \\mid D\\mbox{ diagonal}\\}$.\n\nAlternatively, we can use the Wasserstein 2-distance\n\\cite{olkin1982,ning2011} between the respective Gaussian\nprobability density functions, which can be written in the form of a\nsemidefinite program\n\\[\nd(\\hat\\Sigma+D, \\Sigma)=\\min_{C_1}\\left({\\operatorname{trace}}(\\Sigma+\\hat\\Sigma+D+C_1+C_1') \\mid \\left[\n\\begin{array}{cc} \\hat\\Sigma+D & C_1 \\\\\n C_1' & \\Sigma \\\\\n\\end{array}\n\\right]\\geq0 \\right).\n\\]\n\nReturning to the uncertainty radius of Section \\ref{correspondence} and the\nproblem discussed in Section \\ref{sec:min-max}, we note that the problem\n\\begin{equation}\\nonumber\n\\max\\min_{K} L(K, \\hat \\Sigma,D)\\\\\n=\\max {\\operatorname{trace}}\\left(\\hat\\Sigma-\\hat\\Sigma(\\hat\\Sigma+D)^{-1}\\hat\\Sigma\\right)\n\\end{equation}\ncan be expressed as the semidefinite program\n\\begin{equation}\\nonumber\n\\max_Q \\left\\{ {\\operatorname{trace}}\\left(\\hat\\Sigma-Q \\right)\\mid\n\\left[\n \\begin{array}{cc}\n Q & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\hat\\Sigma+D \\\\\n \\end{array}\n \\right]\\geq 0\n\\right\\}.\n\\end{equation}\nThus, putting the above together, a formulation that incorporates the various tradeoffs between the dimension of the signal subspace, mean-square-error loss, and statistical errors is to maximize\n\\begin{equation}\\label{eq:maxmin2}\n{\\operatorname{trace}}(\\hat\\Sigma -Q) - \\lambda_1\\, {\\operatorname{trace}}(\\hat\\Sigma) -\\lambda_2\\, {\\operatorname{trace}}(\\hat\\Sigma + D - C_1-C_1^\\prime)\n\\end{equation}\nsubject to\n\\begin{eqnarray*}\n\\left[\n \\begin{array}{cc}\n Q & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\hat\\Sigma+D \\\\\n \\end{array}\n \\right]\\geq 0,\\;\\left[\n\\begin{array}{cc} \\hat\\Sigma+D & C_1 \\\\\n C_1' & \\Sigma \\\\\n\\end{array}\n\\right]\\geq0, \\mbox{ with }D\\geq 0 \\mbox{ and diagonal.}\n\\end{eqnarray*}\nThe value of the parameters $\\lambda_1$, $\\lambda_2$ dictate the relative importance that we place on the various terms and determine the tradeoffs in the problem.\n\nWe conclude with an example to highlight the potential and limitations of the techniques.\nWe generate data $X$ in the form\n\\[\nX=FV+\\tilde X\n\\]\nwhere $F\\in{\\mathbb R}^{n\\times r}$, $V\\in {\\mathbb R}^{r\\times T}$, and $\\tilde X\\in {\\mathbb R}^{n\\times T}$ with $n=50$, $r=10$, $T=100$. The elements of $F$ and $V$ are generated from normal distributions with mean zero and unit covariance. The columns of $\\tilde X$ are generated from a normal distribution with mean zero and diagonal covariance, itself having (diagonal) entries which are uniformly drawn from interval $[1, 10]$. The matrix $\\Sigma=XX'$ is subsequently scaled so that ${\\operatorname{trace}}(\\Sigma)=1$.\nWe determine\n\\[\n(\\hat\\Sigma,Q,D)={\\rm arg}\\max \\left\\{ {\\operatorname{trace}}(\\hat\\Sigma-Q)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\right\\}\n\\]\nsubject to\n\\begin{eqnarray*}\n\\left[\\begin{matrix}Q &\\hat\\Sigma\\\\ \\hat\\Sigma & \\hat\\Sigma+D \\end{matrix} \\right]\\geq0, ~d(\\hat\\Sigma+D, \\Sigma)\\leq \\epsilon, \\text{~with~} \\hat\\Sigma, D\\geq0 \\text{~and~} D \\text{~diagonal},\n\\end{eqnarray*}\nand tabulate below a typical set of values for the rank of $\\hat\\Sigma$ (Table 1)\nas a function of $\\lambda$ and $\\epsilon$. We observe a ``plateau'' where the rank stabilizes at $10$ over a small range of values for $\\epsilon$ and $\\lambda$. Naturally, such a plateau may be taken as an indication of a suitable range of parameters.\nAlthough the current setting where a small perturbation in the empirical covariance $\\Sigma$ is allowed, the bounds for the rank\nin \\eqref{Guttman:bound2} and \\eqref{eq:lowerbound3} are still pertinent. In fact, for this example, in $7\/10$ instances where the ${\\operatorname{rank}}(\\hat\\Sigma)=10$ the bound in \\eqref{Guttman:bound2} (computed based on the perturbed covariance $\\hat\\Sigma+D$) has been tight and it thus a valid certificate. For the same range of parameters, the bound in \\eqref{eq:lowerbound3} has been lower than the actual rank of $\\hat\\Sigma$. In general, the bounds in \\eqref{Guttman:bound2} and \\eqref{eq:lowerbound3} are not comparable as either one may be tighter than the other.\\\\\n\\begin{center}\n\\begin{minipage}[]{.6\\textwidth}\n\\begin{tabular}[t]{|c||c|c|c|c|c|c|c|}\n \\hline\n \t\\backslashbox[.5cm]{$\\lambda$}{$\\epsilon$}\t& $0$& $0.08$ & $0.10$& $0.12$& $0.14$ & $0.16$\\\\ \\hline\\hline\n $1$ & 46 & 26 & 24 & 23 & 22 & 22 \\\\ \\hline\n $5$ & 46 & 17 & 14 & 10 & 10 & 9 \\\\ \\hline\n $10$ & 45 & 16 & 12 & 10 & 10 & 8 \\\\ \\hline\n $20$ & 45 & 15 & 12 & 10 & 10 & 8 \\\\ \\hline\n $50$ & 45 & 15 & 12 & 10 & 10 & 8 \\\\ \\hline\n $100$& 45 & 15 & 11 & 10 & 10 & 8 \\\\ \\hline\n\\end{tabular}\\\\[.05in]\n{Table 1: ${\\operatorname{rank}}(\\hat\\Sigma)$ as a function of $\\lambda$ and $\\epsilon$}\\\\[.05in]\n\\end{minipage}\n\\end{center}\n\n\\section{Conclusions} \\label{sec:conclusion}\n\nIn this paper we considered the general problem of identifying\nlinear relations among variables based on noisy measurements --a\nclassical problem of major importance in the current era of ``Big\nData.'' Novel numerical techniques and increasingly powerful\ncomputers have made it possible to successfully treat a number of\nkey issues in this topic in a unified manner. Thus, the goal of the\npaper has been to present and develop in a unified manner key ideas\nof the theory of noise-in-variables linear modeling.\n\nMore specifically, we considered two different viewpoints for the\nlinear model problem under the assumption of independent noise. From\nan estimation viewpoint, we quantify the uncertainty in estimating\n``noise-free'' data based on noise-in-variables linear models. We\nproposed a min-max estimation problem which aims at a uniformly\noptimal estimator --the solution can be obtained using convex\noptimization. From the modeling viewpoint, we also derived several\nclassical results for the Frisch problem that asks for the maximum\nnumber of simultaneous linear relations. Our results provide a\ngeometric insight to the Reiers\\o l theorem, a\n generalization to complex-valued matrices, an\niterative re-weighting trace minimization scheme for obtaining\nsolutions of low rank along with a characterization of fixed points,\nand certain computational tractable lower bounds to serve as\ncertificates for identifying the minimum rank. Finally, we consider\nregularized min-max estimation problems which integrate various\nobjectives (low-rank, minimal worst-case estimation error) and\nexplain their effectiveness in a numerical example.\n\nIn recent years, techniques such as the ones presented in this work\nare becoming increasingly important in subjects where one has very\nlarge noisy datasets including medical imaging, genomics\/proteomics,\nand finance. It is our hope that the material we presented in this\npaper will be used in these topics. It must be noted that throughout\nthe present work we emphasized independence of noise in individual\nvariables. Evidently, more general and versatile structures for the\nnoise statistics can be treated in a similar manner, and these may\nbecome important when dealing with large databases.\n\nA very important topic for future research is that of dealing with\nstatistical errors in estimating empirical statistics. It is common\nto quantify distances using standard matrix norms --as is done in\nthe present paper as well. Alternative distance measures such as the\nWasserstein distance mentioned in Section~\\ref{statisticalerrors}\nand others (see e.g., \\cite{ning2011}) may become increasingly\nimportant in quantifying statistical uncertainty.\n\nFinally, we raise the question of the asymptotic performance of certificates such as those presented in Section \\ref{sec:CertifMinRank}. It is important to know how the tightness of the certificate to the minimal rank of linear models relates to the size of the problem.\n\n\n\\section*{Acknowledgments}\n\nThis work was supported in part by grants from NSF, NIH, AFOSR, ONR,\nand MDA. This work is part of the National Alliance for Medical\nImage Computing (NA-MIC), funded by the National Institutes of\nHealth through the NIH Roadmap for Medical Research, Grant U54\nEB005149. Information on the National Centers for Biomedical\nComputing can be obtained from\nhttp:\/\/nihroadmap.nih.gov \/bioinformatics. Finally, this project was\nsupported by grants from the National Center for Research Resources\n(P41-RR-013218) and the National Institute of Biomedical Imaging\nand Bioengineering (P41-EB-015902) of the National Institutes of\nHealth.\n\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nObscuration of the nuclear \nemission in type~II AGN allows the study of soft X-ray spectral \ncomponents, which are normally outshone by the direct component \nin type~I unobscured objects. It \nhas been well known since the early day of X-ray spectroscopy that \nexcess emission above the extrapolation of the absorbed nuclear \nradiation is present in almost all bright Seyfert~2s (Turner et al. \n1997). This excess appears smooth when measured with instruments with moderate\nenergy resolutions such as CCD. \nHowever, high-resolution (grating) measurements with {\\it Chandra} and \nXMM-Newton revealed that this excess is generally due to a blending of strong \nrecombination lines from He- and H-like transitions of elements from \nCarbon to Nitrogen (Sako et al. 2000, Sambruna et al. 2001, Kinkhabwala \net al. 2002, Armentrout et al. 2007). X-ray spectral diagnostics \n(Kinkhabwala et al. 2002, Guainazzi \\& Bianchi 2006) and a close \nmorphological coincidence between the soft X-rays and the [OIII] in \nExtended Narrow Line Regions (ENLR; Bianchi et al. 2006, Bianchi et al. 2010) strongly \nindicate that the gas is photoionised by the AGN, with an important role \nplayed by resonant scattering. \n\n\nIn this context, NGC~5252 represents an extraordinary laboratory to study the \nfeedback between the AGN output and circumnuclear gas on kpc scale, thanks to \nits spectacular ionisation cones (Tadhunter \\& Tsvetanov 1989).\n\nNGC 5252 is classified as Seyfert 1.9 \n(\\cite{ost}) S0 (\\cite{dev}) nearby (z=0.023,) galaxy (N$_{H,\n Gal}$=2.14$\\times$10$^{+20}$ cm$^{-2}$, Dickey \\& Lockman, 1990). Small\nradio jets (r$\\sim$4'') have\nbeen detected and found to be aligned with the ionisation cones\n(\\cite{wilson94}). Nonetheless, the host galaxy luminosity\n(M$_{R}$$\\sim$-22. \\cite{capetti}), \nmass (M$_{bulge}$$\\sim$2.4$\\times$10$^{11}$M$_{\\sun}$, \\cite{marc}) and the\nmass of the central super-massive black-hole\n(M$_{BH}$$\\sim$10$^{9}$M$_{\\sun}$, \\cite{capetti}) are more typical of \nquasar than Seyfert galaxies. These pieces of evidence led\n\\cite{capetti} to speculate\nthat NGC 5252 is most probably to be considered a QSO relic. This view is in\nagreement with \"downsizing\" scenarios about the evolution of super-massive\nblack-hole (SMBH) in cosmic\ntimes. Accordingly with these scenarios, most massive SMBHs formed and evolved\nearlier than lower mass ones. \n\n\n\n\nIonisation cones are one of the strongest argument in favour of the Seyfert \nunification scenarios (Antonucci 1993). For this reason, NGC~5252 is also an \nimportant laboratory to test AGN geometrical models. From a diferent point of\nview, \nAGN activity has been recognized, since a \nwhile as a key component of the SMBH host galaxy co-evolution and AGN\nfeedback is likely to self-regulate or be responsible of the observed \nproperties (Menci et al. 2004). \nThe very existence of ionization cones witness that feedback\/winds\n are or were active and thus these sources are ideal laboratories\n for feedback.\n\n\nX-ray measurements allows to directly link the \nproperties of the gas emitting optical lines with the intrinsic AGN \npower, which in type~II AGN can be truly measured only at energies \nlarger than the soft photoelectric cut-off due to the AGN obscuring \nmatter. Furthermore, the morphological coincidence between X-rays and \noptical emission in ENLR (Bianchi et al. 2006) points to a fundamental \nphysical link between the two wavebands. They need to be studied \nsimultaneously in order to derive the correct energy budget in the \nionisation cones. Prompted by these motivations, we have performed deep \nX-ray observations of NGC~5252 at the highest spatial and spectral \nresolution currently available with {\\it Chandra} and XMM-Newton. The \nresults of these observations are the subject of this paper.\n\n\n\\section{The nuclear spectrum}\n\nNGC5252 was observed by XMM-Newton on 2003, July 18th, \nwith the EPIC CCDs (MOS and pn in full window, see Tab.~\\ref{tab11}) as the prime instrument\nfor a total duration of $\\sim$67 ks. The\n{\\it Observation Data Files} (ODFs) were reduced and analysed using the \nlatest Science Analysis System (SAS) software package \n(\\cite{gabriel03}) with associated latest \ncalibration files. We used the {\\tt epproc} and {\\tt emproc} tasks to \ngenerate event files and remove dead and hot pixels. \nSeveral time intervals with a high background rate were \nidentified in the\nsingle events light curve at energy $>$10 keV and were \nremoved, yielding a net exposure of $\\sim$50 ks for the MOS\nand $\\sim$38 ks for the pn. Pile-up is negligible in this source,\naccording to the {\\tt epatplot} SAS task outcome.\nPatterns $\\leq$12 and $\\leq$4 were considered for MOS and \npn, respectively. Source counts were extracted from a circular region\nwith radius 50$\\arcsec$, thus encompassing a large\nfraction of the optically-defined galaxy. Background was estimated using both\nblank-sky files and locally from a offset source-free region.\nLight curves in the soft (0.5-2 keV) and hard (2-10 keV) energy bands \nwere first investigated. We found no significant flux nor spectral \nvariations, thus considered the time averaged spectrum.\n\nThe best-fit spectrum is shown in Fig.~\\ref{fig1mc}.\n \\begin{figure}\n \\centering\n \\includegraphics[angle=-90,width=8.0cm]{spe_cont.ps}\n\\vspace{-2.0mm} \\includegraphics[angle=90,width=8.0cm]{0152940101_RGS.ps}\n \\caption{{\\it Upper panel}: XMM-Newton EPIC spectrum extracted from a\n 50$\\arcsec$ region around the NGC~5252 nucleus. For clarity, only pn\n data are presented. {\\it Lower panel:} XMM-Newton RGS spectrum of\n NGC~5252 }\n \\label{fig1mc}\n \\end{figure}\nIf the nuclear emission is modeled with a simple absorbed power-law we\nobtain an extremely flat photon index [$\\Gamma$=1.05$\\pm$0.10 and\n$N_{\\rm H}$=(2.2$\\pm$0.1)$\\times 10^{22}$~cm$^{-2}$, the reported errors\n are, here and hereafter, at 90\\% confidence level], \nplus a soft component emerging at energies below E$\\sim$1 keV. It can be\nparametrised with a scattered power-law with a steep photon index $\\Gamma$=3.0$\\pm$0.2.\nThere is also evidence for\none (or few) soft emission lines in addition to the soft\ncontinuum, a Fe K$_{\\alpha}$ line with \nE$=$6.44$\\pm$0.05 keV and EW=50$\\pm$25 eV, and an absorption edge \nat E=7.0$\\pm$0.1 keV and optical depth $\\tau$=0.31$\\pm$0.05.\n A significantly better fit \nis obtained in case of a power-law plus a thermal\n component, namely $mekal$ in $Xspec$ (\\cite{mewe85}), \n is used to model the soft X-ray band\nof NGC 5252 ($\\chi^{2}$\/d.o.f.=1436\/1410 in the first case while\n$\\chi^{2}$\/d.o.f.=1329\/1410 with $mekal$). \nIn this case, the temperature of the plasma\nis $kT$$\\sim$0.17 keV. \nThe investigation on the true nature of\nthis soft X-ray component will be the subject of the next Sections. \n Here it is important to note that, whatever the fitting of the data below $\\sim$1 keV, the best-fit\nmodel for the nuclear emission of NGC 5252 above $\\sim$1 keV\nis typical in shape, but flatter ($\\Gamma$$\\sim$1.4-1.5) than normally\nfound from Seyfert 2 galaxies ($\\Gamma$$=$1.5-2.5, Turner \\& Pounds 1989; \\cite{turner97,risaliti02,cappi06}; Dadina 2008). It is however \nconsistent with the previous ASCA measurement ($\\Gamma$$\\sim$1-1.5,\n\\cite{cappi96}), confirming the need in this source for a more complex \nabsorption \n(either multiple, ionised or both) in order to recover a steeper \ncanonical photon index.\nWith the above model we measure, for the soft (0.5-2 keV) component, a flux of \n3.5 $\\times$10$^{-13}$erg cm$^{-2}$s$^{-1}$ corresponding to a luminosity of \n4.1$\\times$10$^{41}$ erg\/s and, for the hard (2-10 keV) component, a flux of \n8.9 $\\times$10$^{-12}$erg cm$^{-2}$s$^{-1}$ corresponding to a (unabsorbed) \nluminosity of 1.2$\\times$10$^{43}$ erg\/s. This is consistent, within a few tens\npercent, with previous ASCA and BeppoSAX values (Cappi et al. 1996; Dadina\n2007). It is worth noting here that,\nfrom IR diagnostics, we expect that star forming activity should contribute to\nless than $\\sim$1\\% to the total soft X-ray emission (Cappi et al. 1996). \n\n\n\\section{High-resolution spectroscopy of the AGN environment}\n \\begin{figure*}\n \\centering\n \\includegraphics[width=8cm,angle=-90]{fig2_1.ps}\n \\caption{$Chandra$ ACIS-S images of NGC~5252 in the 0.1--1~keV (soft;\n \t\t{\\it left panel}), and 1--10~keV (hard; {\\it right\n\t\tpanel}) Images were smoothed with a\n\t\t$\\sigma = 1.25$~pixels (yielding an angular resolution of\n $\\simeq$1\" in the images)\n\t\twavelet for illustration purposes. The {\\it\n\t\tthin solid lines} represent\n\t\t9 linearly spaced contours in the range 2 to 20 counts per\n\t\tpixel. The {\\it thick dot-dashed line} indicate the position\n\t\tof the out-of-time events readout streak, removed before\n\t\tthe generation of the image. The {\\it dashed} lines represents\n\t\tthe regions, whence the spectra\n\t\tof the South and North diffuse Spots were\n\t\textracted. The {\\it solid circles} represent the regions,\n\t\twhence the spectra of the nucleus (the brightest and\n central spot) and of the South-East\n\t\tNuclear Source were extracted.\n }\n \\label{fig2}\n \\end{figure*}\n\n\nThe Reflection Grating Spectrometer (RGS; \\cite{derherder01})\non board XMM-Newton\nobserved NGC~5252 simultaneously to the EPIC cameras (Tab.~\\ref{tab11}, Fig.~\\ref{fig1mc}). It\nproduces high-resolution (first order resolution 600-1700~km~s$^{-1}$)\nspectra\nin the 6--35~\\AA\\ (0.35--2~keV) range. Its 2.5$\\arcmin$ diameter\nslit fully encompasses the ionisation cones and the host galaxy.\nThe RGS spectrum represents therefore only the average conditions of\nthe soft X-ray emitting gas across the nucleus and the cone.\n\n\\begin{footnotesize} \n\\begin{table}\n\\caption{Main characteristics of the X-ray observations presented here.} \n\\label{tab11}\n\\centering \n\\begin{tabular}{l c c l c c } \n\\hline\n&&&&&\\\\\nInstr.&Exp.&CR & Instr.&Exp.&CR \\\\\n&&&&&\\\\\n&ks&c\/s&&ks&c\/s\\\\\n&&&&&\\\\\n\\hline\\hline \n&&&&&\\\\\nEpn&38&1.10$^{a}$& RGS1&63& 0.08$^{b}$\\\\\n&&&&&\\\\\nEMOS1&49&0.37$^{a}$ &RGS2&63& 0.08$^{b}$\\\\\n&&&&&\\\\\nEMOS2&50&0.37$^{a}$&ACIS-S&60&0.32$^{a}$\\\\\n&&&&&\\\\\n\\hline\n\\end{tabular}\n\n\n$^{a}$ count-rate in the 0.2-10 keV band; $^{b}$ count-rate in the 0.4-1.2 keV\n\\end{table}\n\\end{footnotesize}\n\n\nRGS data were reduced starting from the {\\it Observation Data Files}\nwith SASv6.5 (\\cite{gabriel03}), and using the latest calibration files. \nThe SAS meta-task {\\tt rgsproc} was used\nto generate source and background spectra, assuming as a reference\ncoordinate coincident with the optical nucleus of NGC~5252. Background\nspectra were generated using both blank field maps - accumulated across\nthe whole mission - and a ``local'' background accumulated during the\nobservation. The former, based on a model of the estimated background on\nthe basis of the count rates detected in the most external of the camera\nCCDs, overestimates the intrinsic background level during the\nobservation. We have therefore employed the ``local'' background hereafter.\nA correction \nfactor to the count background spectrum has been applied to take into \naccount the size of the extraction region, which corresponds to the area \nof the RGS active CCDs outside the 98\\% percentage point of the line \nspread function in the cross-dispersion direction.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{fig3_1.ps}\n \\caption{Radial profile ({\\it filled circles}) of the ACIS-S hard band\n\t\timage. The {\\it solid line}\n\t\trepresents the PSF for a source with the same\n\t\thard X-ray spectral energy distribution as the NGC~5252\n\t\t``nucleus normalized'' to its on-axis peak flux. When not visible,\n the error bars are within the filled circles.\n }\n \\label{fig3}\n \\end{figure}\n\n\nWe simultaneously fit the spectra of the two cameras \nfollowing the procedure outlined in Guainazzi \\& Bianchi\n(2006)\\footnote{This paper discusses a sample of 69 RGS spectra of\ntype 1.5, 1.8, 1.9 and 2 Seyfert galaxies. \nThe observation of NGC~5252 discussed in this paper belongs to this sample as well.}\nwho have performed local spectral fits around\neach of the $\\simeq$40 emission lines detected in the archetypal\nobscured Seyfert NGC~1068 (\\cite{kin02}). In these fits, both the\nbackground level and the continuum have been assumed as independent power-law\ncomponents, with photon index $\\Gamma$ set equal to 1. \nIt is worth noting that the adopted \nvalue of the power law index, here equal to the photon index of the\ncontinuum of the primary emission, does not affect the results signifcantly, given the very\nlimited band of these fits. Different choices for the \ncontinuum spectral index yield indistinguishable results.\nEach emission line has been modeled with an unresolved Gaussian profile \nfixed\nto be at the expected energies (leaving the intrinsic\nwidth of the profile free yields a negligible improvement in the quality of\nthe fit). We detect three lines (see lower panel of Fig. 1) at a confidence level larger than\n90\\% [$\\Delta \\chi^{2} = $ 10.5, 24.0, 10.8 for CV, OVII and OVIII \nlines, respectively, for one interesting parameter (Tab.~\\ref{tab1})].\n\\begin{table}\n\\caption{List of emission lines detected in the RGS spectrum of NGC~5252.\n$E_c$ is the centroid line energy; $L$ is the intrinsic line luminosity:\n$\\Delta v$ is the difference between the measured line centroid energy\nand the laboratory energy. Only statistical error on this measurements\nare quoted. $v_{{\\rm sys}}$ is the systematic error on\n$\\Delta v$ due to residual uncertainties in the RGS aspect solution\n($\\simeq$8~m\\AA)} \n\\begin{footnotesize} \n\\label{tab1} \n\\centering \n\\begin{tabular}{l c c c c} \n\\hline\\hline \nIdentification & $E_c$ & $L$ & $\\Delta v$ & $v_{sys}$ \\\\\n& (eV) & (10$^{40}$~erg~s$^{-1}$) & (km~s$^{-1}$) & (km~s$^{-1}$) \\\\\n\\hline \nC{\\sc vi} Ly-${\\alpha}$ & $367 \\pm 5$ & $3 \\pm^{11}_{2}$ & $800 \\pm 400$ & 70 \\\\\nO{\\sc vii} He-${\\alpha}$ (f) & $560.4 \\pm 0.2$ & $2.0 \\pm^{0.8}_{0.9}$ & $-370 \\pm 110$ & 110 \\\\\nO{\\sc vii} He-${\\alpha}$ (i) & $E_c (f) + 7.7$ & $<$0.9 & & \\\\\nO{\\sc vii} He-${\\alpha}$ (r) & $E_c (f) + 13.0$ & $<$1.4 & & \\\\\nO{\\sc viii} Ly-${\\alpha}$ & $654.0 \\pm^{0.7}_{1.2}$ & $0.6 \\pm 0.4$ & $300 \\pm^{300}_{600}$ & 130 \\\\\n\\hline \n\\end{tabular}\n\\end{footnotesize}\n\\end{table}\nNone of them is a Radiative Recombination Continuum (RRC). A (admittedly\nloose) constrain on the width of the Gaussian profile can be obtained on\nthe O{\\sc vii} He-$\\alpha$ triplet only: $\\sigma < 4400$~km~s$^{-1}$\n(8.2~eV).\n \\begin{figure*}\n \\centering\n \\hbox{\n \\includegraphics[width=6cm,angle=-90]{fig4a_1.ps}\n \\hspace{1.0cm}\n \\includegraphics[width=6cm,angle=-90]{fig4b_1.ps}\n }\n \\hbox{\n \\includegraphics[width=6cm,angle=-90]{fig4c_1.ps}\n \\hspace{1.0cm}\n \\includegraphics[width=6cm,angle=-90]{fig4d_1.ps}\n }\n \\caption{Spectra ({\\it upper panels}) and residuals in units of\n\t\tstandard deviations ({\\it lower panels}) for the four\n\t\tregions of NGC~5252 defined in Fig.~\\ref{fig2}.\n }\n \\label{fig4}\n \\end{figure*}\nDiagnostic parameters involving the intensity of the O{\\sc vii} He-${\\alpha}$\ntriplets can be in principle used to pinpoint the physical process responsible\nfor the bulk of the X-ray emission in high-resolution spectra. The detection\nof the forbidden ($f$) component only allows to set lower limits\non the standard triplet diagnostics (\\cite{gabriel69,porquet00}):\n$R > 1.1$, $G > 0.7$ (where R is the ratio between forbidden and\nintercobination lines and depends on the electron density, while G is the\nratio between intercombination plus forbidden lines and the resonance line, \\cite{gabriel69,porquet00}). \n\nThese limits, although fully consistent with\nphotoionised plasmas, do not rule out collisional ionisation. Guainazzi \\&\nBianchi (2007) proposed a criterion to discriminate, {\\it on a statistical\nbasis}, between AGN- and starburst-powered sources based on the location\nof the source in an empirical observable plane: integrated luminosity of the\nHe- and H-like Oxygen lines, $L_O$, against the intensity\nratio $\\eta$ between the $f$ and the\nO{\\sc viii} Ly-$\\alpha$. In NGC~5252\n$\\eta = 2.3 \\pm 0.4$, and\n$L_0 \\sim 3 \\times 10^{40}$~erg~s$^{-1}$. These values put NGC~5252 in\nthe plane locus preferentially occupied by photoionised (AGN) sources\n(\\cite{guainazzi09}).\nWe estimated also the flux density associated to the continuum, using a\nline-free energy range between 586 and 606~eV: $\\nu L_{\\nu}|_{0.6 \\ keV} =\n(7.2 \\pm^{1.7}_{2.9}) \\times 10^{40}$~erg~s$^{-1}$.\n\n\\section{X-ray imaging of the ionisation cones}\n\n{\\it Chandra} observed NGC~5252 on August 11, 2003 with the ACIS-S\ndetector in standard VFAINT configuration. Data reduction was performed\nwith CIAO version 3.3 and associated CALDB version 3.2. ``Level~1''\nevents were corrected for bad pixels, gain spatial dependency,\nand charge transfer inefficiency via {\\tt acis\\_process\\_events}.\n\nAlthough the correction for read-out streaks was applied\nas well, some out-of-time events remain in the final cleaned event\nlist, and were removed by applying a 2 pixels ($\\simeq$1$\\arcsec$)-wide\ntilted rectangular box around the streak.\n\nACIS-S images in the\n$\\sim$2$\\arcmin$ around the optical core of\nNGC~5252 are shown in Fig.~\\ref{fig2} in the 0.2--1~keV and\nin the 1--10~keV energy bands. The soft band clearly shows extended emission\nin the North-South direction on both sides of the nucleus. On the\ncontrary, the hard band image is point-like. We extracted a radial\nprofile of the latter, and compared it with the expected instrumental Point\nSpread Function (PSF) of a source\nwith the same spectral energy distribution as the\nNGC~5252 nucleus. The two profiles are perfectly consistent\nup to 30$\\arcsec$ off-axis (see Fig.~\\ref{fig3}).\n\nIn order to characterize the spectral behavior of the diffuse emission,\nwe have extracted spectra from four regions, identified in the soft image\n(Fig.~\\ref{fig2}): the nucleus (N), a S-E source about 3.2$\\arcsec$ from the\nnucleus (SENS), and the South (SS) and North (NS) diffuse Spots. Background\nspectra were generated from a large circle 57$\\arcsec$ wide around the\ngalaxy core, once a $21$$\\arcsec$ inner circle, as well as\n5$\\arcsec$ circles around each serendipitous point sources were removed. Alternative\nchoices of the background regions do not substantially change\nthe results presented in this section. Source spectra were rebinned\nin order to over-sample the intrinsic instrumental energy resolution by\na factor $\\ge$3, and to have at least 25 background-subtracted counts\nin each spectral bin. The latter criterion\nensures the applicability of the $\\chi^2$ statistics.\n\nFor all spectra, we have employed a baseline model that include \na thermal emission component from collisionally excited plasma ({\\tt mekal} in\n{\\sc Xspec}; \\cite{mewe85}). \nThis choice was done for simplicity and the only information\nobtained using {\\tt mekal} is the flux of the thermal component. \nThis is particularly true for the SS and NS regions where the complexity of \nthe {\\tt mekal} model is well above the quality of the data. \nMoreover, a photoelectrically-absorbed power-law was\nalways included in the data. \nThe physical meaning of the latter is different\ndepending on the region where the spectrum was extracted. For the nuclear\nregion, the non-thermal component represents the contribution of the active\nnucleus; for the other regions, the integrated contribution of hard galactic\nsources such as, for example, X-ray binaries, cataclysmic variables or \nsupernova remnants. We therefore\nrefrain from attributing a physical meaning to the power-law spectral\nindeces in the latter case.\nThe spectra and corresponding best-fits are shown in Fig.~\\ref{fig4}.\nA summary of the spectral results is presented\nin Tab.~\\ref{tab2}. The {\\it Chandra} data confirm that the nuclear spectrum\n\\begin{table*}\n\\label{tab2}\n\\caption{ACIS-S best-fit parameters and results for the spatially-resolved regions\nof NGC~5252. $E_c$ and $EW$ are the centroid energy and the Equivalent Width\nof a Gaussian profile at the energies of K$_{\\alpha}$ fluorescence for\nneutral or mildly ionised iron. Fluxes, ($F$), are in the observed frame.\nLuminosities, ($L$), are in the source frame, and are corrected for absorption.}\n\\begin{tabular}{lcccccccccc} \\hline \\hline\nRegion & $N_H$ & $\\Gamma$ & $kT^a$ & $E_c$ & $EW$ & $F_{0.5-2 keV}$$^b$ &\n$F_{2-10 keV}$$^b$ & $L_{0.5-2 keV}$$^c$ & $L_{2-10 keV}$$^c$ & $\\chi^2\/\\nu$ \\\\\n& $(10^{22}$~cm$^{-2}$) & & (eV) & (keV) & (eV) & & & & & \\\\ \\hline\nNucleus & $2.32 \\pm^{0.13}_{0.15}$ & $1.00 \\pm^{0.08}_{0.06}$ & $140\\pm^{60}_{40}$ & $6.37 \\pm^{0.06}_{0.05}$ & $50 \\pm 20$ & $2.66\\pm^{0.11}_{0.28}$ & $103\\pm^2_3$ & $0.82\\pm^{0.03}_{0.09}$ & $1130 \\pm^{20}_{30}$ & 176.9\/127 \\\\\nSENS & $0.47\\pm^{0.16}_{0.31}$ & $0.8 \\pm 0.4$ & $<140$ & ... & ... & $0.06 \\pm 0.05$ & $0.34 \\pm^{0.08}_{0.14}$ & $50 \\pm 40$ & $4.1 \\pm^{1.0}_{1.7}$ & 3.0\/5 \\\\\nNS & $\\equiv$$N_{H,Gal}$ & $-0.9\\pm^{0.5}_{0.6}$ & $240 \\pm^{40}_{30}$ & ... & ... & $0.066\\pm^{0.011}_{0.015}$ & $1.3 \\pm 0.3$ & $0.89 \\pm^{0.15}_{0.20}$ & $15 \\pm 3$ & 1.0\/3 \\\\\nSS & $\\equiv$$N_{H,Gal}$ & $0.0 \\pm 0.4$ & $110\\pm^{13}_{21}$ & ... & ... & $0.122 \\pm^{0.017}_{0.023}$ & $0.70 \\pm^{0.18}_{0.53}$ & $1.7 \\pm^{0.2}_{0.3}$ & $8 \\pm^2_6$ & 5.1\/7 \\\\\n& & & $510\\pm^{170}_{230}$ & & & & & & & \\\\ \\hline\n\\end{tabular}\n\n\\noindent\n$^a$derived using $mekal$ to fit the spatially resolved data\n\n\\noindent\n$^b$in units of $10^{-13}$~erg~s$^{-1}$~cm$^{-2}$\n\n\\noindent\n$^c$in units of $10^{40}$~erg~s$^{-1}$\n\n\\end{table*}\nis remarkably flat, and is\nseen through a substantial column density ($N_H \\simeq\n2.26 \\times 10^{22}$~cm$^{-2}$).\n\nThe background subtraction for the spectrum of SENS could be contaminated\nby the spilling of the nuclear emission.\nThe encircled energy fraction at a distance equal to that between\nsources N and SENS is $\\simeq$97.5\\%. However,\nsubtracting a properly rescaled nuclear spectrum to the SENS spectrum yields\nnegative counts above 2~keV. In order to have an independent estimate of the\nspectral behavior of SENS, we have extracted images\n10$\\arcsec$ around the nuclear region in narrow energy bands\n(Fig.~\\ref{fig5}): 200-400~eV,\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{fig5_1.ps}\n \\caption{Narrow-band ACIS-S images in the 10$\\arcsec$ around the\n\t\tNGC~5252 core, normalized to the peak nuclear emission.\n\t\tImages are smoothed with a 5 pixel Gaussian kernel.\n\t\tThe {\\it solid lines} in the {\\it upper left} panel\n\t\trepresent a contour plot of the 0.2--1~keV \n\t\timage, assuming the same smoothing criterion.\n }\n \\label{fig5}\n \\end{figure}\n500-600~eV, 600-700~eV and 700-1000~eV. In a line-dominated plasma, the\nabove energy ranges correspond to bands dominated by C{\\sc vi}\nand C{\\sc v} K$_{\\alpha}$, O{\\sc vii} He-$\\alpha$, O{\\sc viii} Ly-$\\alpha$,\nand Fe-L transitions, respectively.\nEach image was normalized to the peak of the\nnuclear emission in that energy band. The soft X-ray\nSENS spectrum is comparatively dominated by Oxygen transitions, with little\ncontribution in either the Carbon or the Iron band.\n\n\n\\section{Comparing soft X-ray and [OIII] morphologies}\n\nNGC 5252 was observed in the [OIII] band with the WFPC2 on-board HST on\n1995, July 23, using the linear ramp filter FR533N. The data were downloaded\nfrom MAST (multi-mission archive at STScI). The images were processed through the standard OTFR (on-the-fly reprocessing) calibration pipeline which performs analog-to-digital conversion, bad pixel masking, bias and dark subtraction, flat field correction and photometric calibration. The cosmic rays rejection was performed combining the two images that are usually taken for this scope. Geometric distortion was corrected using the {\\it multi drizzle} script (\\cite{koekemoer02}).\n\nThe relative Chandra-HST astrometry is clearly a fundamental issue for this\nwork. Chandra has a nominal position accuracy of 0.6$\\arcsec$ while the\nabsolute astrometry of HST is accurate to 1-2$\\arcsec$. Fortunately, to align\nthe two astrometric solutions, we could use a point-like source detected both\nin the WFPC2 and Chandra fields. This source was previously detected at radio\nwavelengths (\\cite{wilson94}) and is most probably associated to a background\nquasar (Tsvetanov et al. 1996). Moreover, as a second reference point, we used the brightest emission peak in the nuclear region of NGC 5252 itself.\n\n\nImages were calibrated in flux using a constant flux conversion\nfactor of $1.839 \\times 10^{-16}$, corresponding to the flux producing a\ncount rate of 1~s$^{-1}$ in the filter band. The above value is appropriate\nfor the instrumental configuration employed during the \nNGC~5252 exposure,\nas indicated by the {\\tt PHOTFLAM} keyword in the image file.\n\nThe HST image is shown in Fig.~\\ref{fig6}.\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{fig6_22.ps}\n \\caption{HST WFC~2 [OIII] image of the ionisation cones in NGC~5252.\n }\n \\label{fig6}\n \\end{figure}\nTsvetanov et al. (1996), Morse et al. (1998) \\& Capetti et al. (2005)\ndiscussed it in details.\nWe refer the reader to these papers for an extensive\ndiscussion on the properties of the optical emission. Their main\noutcomes can be summarized as follows:\n\n\\begin{itemize}\n\n\\item the surface brightness is dominated by the unresolved nucleus\n\n\\item a half-ring structure is apparent S-E of the nucleus at a\nmaximum projected distance of $\\simeq$1.5~kpc. It is probably associated\nwith the near side of an inclined gas disk, whose far side is\nobscured by the host galaxy dust (\\cite{morse98});\n\n\\item the large scale ionisation cone is traced by thin shells of\nenhanced emission at either side of the nucleus, well aligned along\na P.A.$\\simeq$110$^{\\circ}$ at distances between 5 and 11~kpc. Fainter\nco-aligned structures at scales as large as 20~kpc are \ndetected as well in the O[{\\sc iii}] images;\nhowever, we will not discuss these latter structures, as\nthey are beyond the region where X-ray emission associated with\nNGC~5252 is detected;\n\n\\item there is no evidence of radial motions. The measured velocities\nof the different structures are fully explained by the rotations of the \ntwo disks [nonetheless Acosta-Pulido et al. (1996) claimed the detection\nof radial motions describing the kinematic properties of the [OIII] emission \narcs].\n\n\\end{itemize}\n\nIn Fig.~\\ref{fig7} we present\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm,angle=-90]{fig7.ps}\n \\caption{Iso-intensity {\\it Chandra}-ACIS 0.2-1 keV X-ray iso-intensity\n \t\tcontours superposed to the HST WFC~2 [OIII] image\n \t\tof Fig.~\\ref{fig7}. The resolution of the latter has been\n \t\tdegraded to the typical resolution of {\\it Chandra} optics\n \t\tby applying a wavelet smoothing with an 8~pixel\n \t\tkernel. The {\\it Chandra} contours represents\n \t\tnine linearly spaces count levels from 0.5 to 20 counts\n \t\tper pixel, after a wavelet smoothing with a $\\sigma$=1.25\n \t\tpixel has been applied.\n }\n \\label{fig7}\n \\end{figure}\nthe superposition between the soft X-rays iso-intensity contours to\nthe [OIII] image. The HST image spatial resolution has been\ndegraded with a 8~pixel wavelet kernel to match the resolution\nof the {\\it Chandra} optics.\nRegions of enhanced X-ray emission exhibit a remarkable coincidence with the\nmorphology of the optical narrow-band image. \n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=5.5cm,angle=-90]{fig91.ps}\n \\caption{[OIII]\/Soft-X flux ratio as a function of distance from the nucleus}\n \\label{fig8}\n \\end{figure}\n\n\nWe have calculated the ratio between the\n[OIII] band and the 0.5-2 keV flux (Fig. 8) for the regions specified, \nafter splitting region SS into two\nsub-regions divided by a E-W line at $\\delta_{J2000} = 4^{\\circ} 32\\arcmin\n21\\arcsec$ (regions ``SSNorth'' - SSN - and ``SSSouth'' - SSS - respectively).\nThe ratio exhibit a dynamical range smaller than a factor 2\nover distances ranging from less than 100 pc\nto $\\sim$1.5 kpc with a slight tendency to decrease with the distance from the \nnucleus(r). This last effect, however, is most probably an observational \nartifact due to the decreasing in surface brightness of the arcs moving away\nfrom the nucleus coupled with the sensitivity limits of $Chandra$ to extended\nsources. At a first glance, the [OIII]\/Soft-X ratio profile as a function of the \ndistance from the nucleus seems to \nsuggest that the electron density follow a r$^{-2}$ relation since the number \nof ionising photons and of the overall average ionisation state of the \nnuclear species remain almost constant.\nThis result is in \nagreement with Bianchi et al. (2006), who assumed, however,\na very simplified geometry of the emitting gas. \nA more detailed investigation on this\ntopic in NGC~5252 is hampered by the quality of the data.\n\n\n\\section{Discussion}\n\n The soft X-ray emission of NGC~5252 is clearly\n extended and ACIS images demonstrate that \nthe spectacular ionisazion cones observed in [OIII] have counterparts\nin the 0.1-1 keV band. \nThe cumulative soft X-ray spectrum observed by $XMM$--$Newton$ is described by\na soft power-law ($\\Gamma$$\\sim$3). The ACIS images suggest that this is probably due \nto a blend of emission lines that mimics such steep power-law as demonstrated \nin other type II Seyferts like NGC 1068, Circinus galaxy and Mrk 3 (\\cite{kin02}, \\cite{brink02},\n\\cite{ogle03}, \\cite{sam01}, \\cite{sako00}, \\cite{b05}, \\cite{pp05}).\nThis scenario is supported also by the detection in the RGS high resolution \nspectrum of three emission \nlines, of CV, OVII and OVIII, probably due to photoionised gas. This is consistent also by previous \noptical studies that excluded \ncollisional ionisation along the cones of NGC~5252 (Tsvetanov et al. 1996). \nMoreover, the presence of {\\it in situ} ionisation sources due to shocks formed by \nlarge scale outflows interacting with the interstellar matter has been\nexcluded (\\cite{morse98}). \nThus the source of ionising photons is most probably the nucleus. Under this\nassumption, \nwe can use the imaging of the arcs to study the physical condition\nof the gas along the ionisation cones. In particular, \nthe constant of [OIII]\/(0.5-2 keV) flux ratio along the ionisation cones \nwithin the inner\\footnote{The outer arcs and filaments (\\cite{tad}) are most probably too weak \nto be detected in X-rays. Considering the extension of the outer [OIII] arcs, the\nminimum detectable flux between 0.1-1 keV is F$_{0.1-1\n keV}$$\\sim$5$\\times$10$^{-15}$erg s$^{-1}$ cm$^{-2}$ \nwhile, assuming\na constant [OIII]\/soft X-ray ratio, the expected flux should be $\\sim$10 times\nlower. } $\\sim$1.5 kpc suggests a r$^{-2}$ law\nfor the ion density. \n\n\n\n\n\n\n\n\n\nOptical spectroscopic studies (Acosta-Pulido et al. 1996) suggest that the\nradial dependency of the ionization parameter\\footnote{U=$\\frac{L}{4 \\pi \\rho\n r^{2}}$, where L is the source's luminosity,\n $\\rho$ is the density of the ionized gas, $r$ is the distance between the\n source of ionizing photons and the ionized matter.} U follows a different law\nin the south-east ($U \\propto r^{-0.4}$) with respect to the north-east\n($U \\propto r^0$) cone. The authors speculate that the intrinsic behavior should be the one\nshown in the former, while the radial-independence of the ionization\nparameter in the latter may be due to a ``conspiracy'' introduced by the\nexistence of two counterotating disks \nof gas (\\cite{morse98}): one is coplanar \nto the stellar one, and another is inclined by $\\sim$40$^{\\circ}$. \nMorse et al. (1998) speculated that the prominence of the southeast [OIII] \ncone in the nuclear regions is due to the fact that this component is \nseen directly, while the northeast [OIII] cone is seen\nthrough the gas of the other disk. \nIf so, the absorption due to this component could alter the line ratios \npresented by \\cite{ap96} and thus the correct behavior of U should be the one derived from\nthe south-east cone. U$\\propto$r$^{-0.4}$ implies that the luminosity of the nucleus increased by a\nfactor $\\Delta$L$\\sim$3-6 in the last $\\Delta$t$\\sim$5000 years. These numbers\nbecome $\\Delta$L$\\sim$10-30 and $\\Delta$t$\\sim$30000 years if we further \nassume that\nthe U and the ion density laws are still valid up to 10 kpc from the nucleus,\ni.e. where the optical cones are still detectable in [OIII] but not in\nX-rays. \n\n\n\n\n\n\nOn the contrary, having U constant and $\\rho \\propto$ r$^{-2}$ would imply that L has remained constant during the last 5000\n(30000) years. This is consistent with the ``quasar-relic'' scenario proposed by\n\\cite{capetti}. These authors suggested that the nucleus of NGC~5252 is indeed\nthe ``relic'' of a nucleus that already experienced the activity phase in the\npast and that now persists in an almost quiescent phase. \nThis is suggested by the high mass of the SMBH (M$_{BH}$=10$^{9}$M$_{\\sun}$, \nCapetti et al. 2005) that \nindicates that the nucleus has already accreted in the past, the low \nEddington ratio (L$\\sim$10$^{-3}$L$_{Edd}$, assuming the bolometric\ncorrection from Marconi et al. (2004),\nL$_{hard-x}$$\\sim$(1\/22)$\\times$L$_{bol}$), and the early type (S0) morphology\nof the \nAGN host galaxy. In literature it is also reported that the optical emission\nline ratios in the inner 30\" are typical of LINERS \n (\\cite{gon}), thus suggesting that a low efficiency\nengine is acting at the nucleus of the source. \nIt is worth noting that also the detection of two counterotating\ndisks suggests that NGC~5252 is a \"quasar-relic\". These\ndisks are tracers of a major merging event that occurred, most probably, \nmore than 10$^8$ years ago, since the stellar disk of NGC~5252 is\nundisturbed. If the merging event triggered a phase of AGN \nactivity (see Jogge 2006, and references therein for a discussion on this\ntopic), we can expect that it lasted few\/some \n$\\sim$10$^{7}$ years (\\cite{mar}; \\cite{ste}; \\cite{jac}; \\cite{gon2}) \n after which the source has persisted in a quiescent state.\n\n\nFinally, it is worth noting that the spectrum of the nucleus hosted by\nNGC~5252 is confirmed to be quite flat (Cappi et al. 1996). \nAs shown, if modeled with a simple absorbed power-law its photon\nindex points to a very hard spectrum ($\\Gamma$$\\sim$1). \nThe low EW ($\\sim$50 eV) of the neutral (E$_{FeK\\alpha}$$\\sim$6.4 keV) iron line\nis consistent with what expected if the FeK$\\alpha$ line is\nproduced via transmission in the observed column\n(N$_{H}$$\\sim$2$\\times$10$^{22}$ cm$^{-2}$, Makishima 1986) thus excluding a reflection\ndominated spectrum.\nTo reconcile, at least marginally, \nthe hardness of the NGC~5252 nuclear spectrum, with previous results for\nSeyfert galaxies ($\\Gamma$$\\sim$1.5-2.5; Turner \\& Pounds, \n1989; Nandra \\& Pounds, 1994; Smith \\& Done, 1996; Dadina 2008), we must invoke\ncomplex absorption models involving partial covering of the source and\/or \nthe presence of ionised absorbers along the line of sight. In this\ncase the spectral index becomes $\\Gamma$$\\sim$1.4-1.5. \n It is interesting to note that the flat photon index \nmay be a further clue suggesting that the X-rays may be \nproduced in an ADAF,\nNarayan \\& Yi, 1994) as expected in a ``quasar-relic''.\n\n\n\n\n\n\\begin{acknowledgements}\n\nThis paper is based on observations obtained with XMM-Newton, an ESA\nscience mission with instruments and contributions directly funded by\nESA Member States and the USA (NASA). MD greatfully acknowledge Barbara De\nMarco for the helpful discussions. MD, MC and GM greatfully acknowledge \nASI financial support under contract I\/23\/05\/0. CV greatfully acknowledge \nASI financial support under contract I\/088\/06\/0.\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Background}\n\\subsubsection*{Philosophy}\nFor us, \\emph{Real spectrum} is a loose term for a $C_2$-spectrum built\nfrom the $C_2$-spectrum $M\\R$ of Real bordism, considered by Araki,\nLandweber and Hu--Kriz \\cite{HK}. The present article shows that\nbringing together Real spectra and Gorenstein duality reveals rich and interesting structures. \n\nIt is part of our philosophy that theorems about Real spectra can\noften be shown in the same style as theorems for the underlying\ncomplex oriented spectra although the details might be more\ndifficult, and groups needed to be graded over the real representation\nring $RO(C_2)$ (indicated by $\\bigstar$) rather than over the integers\n(indicated by $*$). This extends a well known\nphenomenon: complex orientability of equivariant spectra\nmakes it easy to reduce questions to integer gradings, and we show \nthat even in the absence of complex orientability, good\nbehaviour of coefficients can be seen by grading with\nrepresentations. \n\n\\subsubsection*{Bordism with reality}\nIn studying these spectra, the real regular representation $\\rho=\\R\nC_2$ plays a special role. We write $\\sigma$ for the sign\nrepresentation on $\\R $ so that $\\rho =1+\\sigma$. \n One of the crucial features of $M\\R$ is that \n it is \\emph{strongly even} in the sense of \\cite{HM}, i.e.\n\\begin{enumerate}\n\\item \\label{item:Restr} the restriction functor $\\pi_{k\\rho}^{C_2}M\\R \\to \\pi_{2k}MU$ is an isomorphism for all $k\\in\\Z$, and\n\\item the groups $\\pi_{k\\rho-1}^{C_2}M\\R$ are zero for all $k\\in\\Z$.\n\\end{enumerate}\n\nWe now localize at 2, and (with two exceptions) all spectra and\nabelian groups will henceforth be $2$-local. The Quillen idempotent\nhas a $C_2$-equivariant refinement, and this defines the\n$C_2$-spectrum $BP\\R$ as a summand of $M\\R_{(2)}$. The isomorphism (\\ref{item:Restr}) allows us to lift the usual $v_i$ to classes $\\vb_i\\in\\pi_{(2^i-1)\\rho}^{C_2}BP\\R$. The Real spectra we are interested in are quotients of $BP\\R$ by sequences of $\\vb_i$ and localizations thereof. For example, we can follow \\cite{HK} and \\cite{Hu} and define \n$$BP\\R \\langle n \\rangle = BP\\R\/(\\vb_{n+1},\\vb_{n+2},\\dots)$$ \nand \n$$E\\R(n) = BP\\R \\langle n \\rangle[\\vb_n^{-1}].$$\nThese spectra are still strongly even, as we will show. Apart from the big literature on $K$-theory with Reality (e.g.\\ \\cite{Ati66}, \\cite{Dugger} and \\cite{B-G10}), Real spectra have been studied by Hu and Kriz, in a series of papers by Kitchloo and Wilson (see e.g.\\ \\cite{K-W15} for one of the latest installments), by Banerjee \\cite{Banerjee}, by Ricka \\cite{Ricka} and by Lorman \\cite{Lorman}. \n A crucial point is that a morphism between strongly even\n $C_2$-spectra is an equivalence if it is an equivalence of underlying\n spectra \\cite[Lemma 3.4]{HM}. \n\nWe are interested in two dualities for Real spectra: Anderson duality\nand Gorenstein duality. These are closely related \\cite{GS} but apply\nto different classes of spectra. \n\n\\subsubsection*{Anderson duality}\nThe Anderson dual $\\Z^X$ of a spectrum $X$ is an integral version of\nits Brown-Comenetz dual (in accordance with our general principle,\n$\\Z$ denotes the $2$-local integers). The homotopy groups of the\nAnderson dual lie in a short exact sequence\n\\begin{align}\\label{eq:IntroductionSES} 0 \\to \\mathrm{Ext}_{\\Z}^1(\\pi_{-*-1}X ,\\Z) \\to \\pi_*(\\Z^X) \\to\n\\mathrm{Hom}_{\\Z}(\\pi_{-*}X, \\Z) \\to 0. \\end{align}\n\nOne reason to be interested in the computation of Anderson duals is\nthat they show up in universal coefficient sequences (see\n\\cite{Anderson} or Section \\ref{subsec:Anderson}). The situation is\nnicest for spectra that are Anderson self-dual in the sense that\n$\\Z^X$ \nis a suspension of $X$. Many important spectra like $KU$, $KO$, $Tmf$ \\cite{Sto12} or\n$Tmf_1(3)$ are indeed Anderson self-dual. These examples are all unbounded as the sequence \\eqref{eq:IntroductionSES} nearly forces them to be. \n\nAnderson duality also works $C_2$-equivariantly\nas first explored by \\cite{Ricka}; the only change in the above short\nexact sequence is that equivariant homotopy groups are used. The $C_2$-spectra $K\\R$\n\\cite{H-S14} and $Tmf_1(3)$ \\cite{HM} are also $C_2$-equivariantly\nself-Anderson dual, at least if we allow suspensions by\n\\emph{representation spheres}. \n\nOne simpler example is essential background: if $\\underline{\\Z}$ denotes the constant Mackey functor (i.e., with restriction being the identity and\ninduction being multiplication by 2) then the Anderson dual of its\nEilenberg-MacLane spectrum is the Eilenberg-MacLane spectrum for the\ndual Mackey functor $\\underline{\\Z}^*=\\mathrm{Hom}_{\\Z}(\\underline{\\Z}, \\Z)$\n(i.e., with restriction being multiplication by 2 and induction being\nthe identity). It is then easy to check that in fact $H(\\underline{\\Z}^*)\\simeq \\Sigma^{2(1-\\sigma)}H\\Zu$.\n(From one point of view this is the fact that $\\R P^1=S(2\\sigma)\/\\Ctwo$ is equivalent to the circle). \nThe dualities we find are in a sense all \ndependent on this one. \n\n\\subsubsection*{Gorenstein duality}\nBy contrast with Anderson self-duality, many connective ring spectra\nare Gorenstein in the sense of \\cite{DGI}. We sketch the theory here,\n explaining it more fully in Sections \\ref{sec:kR} and \\ref{sec:dishonest}.\n\nThe starting point is a connective commutative ring $C_2$-spectrum $R$,\nwhose $0$th homotopy Mackey functor is constant at $\\Z$: \n$$\\underline{\\pi}^{\\Ctwo}_0(R)\\cong \\underline{\\Z}.$$ \nThis gives us a map $R\\longrightarrow H\\Zu$ of commutative ring spectra by killing\nhomotopy groups. We say that $R$\nis {\\em Gorenstein} of shift $a\\in RO(C_2)$ if there is an equivalence of $R$-modules\n$$\\mathrm{Hom}_R(H\\Zu ,R)\\simeq \\Sigma^aH\\Zu. $$\n\nWe are interested in the duality this often entails. \nNote that the Anderson dual $\\Z^R$ obviously has the Matlis lifting property\n$$\\mathrm{Hom}_R(H\\Zu, \\Z^R)\\simeq H\\Zu^*, $$\nwhere $\\Z^*=\\mathrm{Hom}_{\\Z}(\\underline{\\Z}, \\Z)$ as above. Thus if $R$ is Gorenstein,\nin view of the \nequivalence $H(\\underline{\\Z}^*)\\simeq \\Sigma^{2(1-\\sigma)}H\\Zu$, we have equivalences\n\\begin{align*}\\mathrm{Hom}_R(H\\Zu ,\\mathrm{Cell}_{H\\Zu} R)&\\simeq \\mathrm{Hom}_R(H\\Zu ,R)\\\\\n&\\simeq \\Sigma^aH\\Zu\\\\\n&\\simeq \\Sigma^{a-2(1-\\sigma)}H(\\underline{\\Z}^*) \\\\\n&\\simeq\n\\mathrm{Hom}_R(H\\Zu, \\Sigma^{a-2(1-\\sigma)}\\Z^R).\\end{align*}\nHere, $\\mathrm{Cell}_{H\\Zu}$ denotes the $H\\Zu$-$\\mathbb{R}$-cellularization as in Section \\ref{sec:Cell}. We would like to remove the $\\mathrm{Hom}_R(H\\Zu, \\cdot)$ from the composite equivalence above. \n\n\\begin{defn}\nWe say that $R$ has {\\em Gorenstein duality} of shift $b$ if we have\nan equivalence of $R$-modules\n$$\\mathrm{Cell}_{H\\Zu} R \\simeq \\Sigma^b \\Z^R.$$\n\\end{defn}\n\n As in the non-equivariant setting, the passage from Gorenstein to\n Gorenstein duality requires showing that\nthe above composite equivalence is compatible with the right \naction of $\\cE =\\mathrm{Hom}_R(H\\Zu, H\\Zu)$. This turns out to be considerably\nmore delicate than the non-equivariant counterpart because\nconnectivity is harder to control; but if one can lift the\n$R$-equivalence to an $\\cE$-equivalence, the conclusion is that if $R$\nis Gorenstein of shift $a$ then it has Gorenstein duality of shift $b=a-2(1-\\sigma)$. \n\n\n\\subsubsection*{Local cohomology}\nThe duality statement becomes more interesting when the cellularization can be\nconstructed algebraically. For any finitely generated ideal $J$ of the $RO(C_2)$-graded\ncoefficient ring $R_{\\bigstar}^{C_2}$, we may form the stable Koszul\ncomplex $\\Gamma_JR$, which only depends on the radical of $J$. In our\nexamples, this applies to the augmentation ideal $J=\\ker(R_{\\bigstar}^{C_2}\\longrightarrow\nH\\Zu_{\\bigstar}^{C_2})$, which may be radically generated by finitely\nmany elements $\\vb_i$ in degrees which are multiples of\n$\\rho$. Adapting the usual proof to the Real context, \nProposition \\ref{prop:cell} shows that \n$\\Gamma_JR\\longrightarrow R $ is\n$H\\Zu$-$\\R$-cellularization: \n$$\\mathrm{Cell}_{H\\Zu}R\\simeq \\Gamma_JR. $$\nThe $RO(C_2)$-graded homotopy groups of $\\Gamma_JR$ can be computed using a spectral sequence involving local cohomology. \n\n\n\n\\subsubsection*{Conclusion}\n In favourable cases the Gorenstein condition on a ring spectrum $R$\n implies Gorenstein duality for $R$; this in turn establishes a strong\n duality property on the $RO(C_2)$-graded coefficient ring, expressed using local cohomology. \n\n\n\n\\subsection{Results}\nOur main theorems establish Gorenstein duality for a large family of Real spectra. We present in this introduction the particular cases of $BP\\R \\langle n \\rangle$ and $E\\R(n)$, deferring the more general theorem to Section \\ref{sec:results}. Let again $\\sigma$ denote the non-trivial\nrepresentation of $\\Ctwo$ on the real line and $\\rho =1+\\sigma$ the \nreal regular representation. Furthermore set $D_n = 2^{n+1}-n-2$ so\nthat $D_n\\rho = |\\vb_1|+\\cdots + |\\vb_n|$. Other terms in the statement will be explained\nin Section \\ref{sec:AKG}. \n \n\\begin{thm}\nFor each $n\\geq 1$ the $\\Ctwo$-spectrum $BP\\R \\langle n \\rangle$ is Gorenstein\nof shift $-D_n\\rho -n$, and has Gorenstein duality of shift\n$-D_n\\rho -n-2(1-\\sigma)$. This means that\n$$\\Z_{(2)}^{BP\\R \\langle n \\rangle} \\simeq \\Sigma^{D_n\\rho+n+2(1-\\sigma)} \\Gamma_{\\Jb_n}BP\\R \\langle n \\rangle,$$\nwhere $\\Jb_n = (\\vb_1,\\dots, \\vb_n)$. This induces a local cohomology\nspectral sequence\n$$H^*_{\\Jb_n}(BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar})\\Rightarrow \\pi^{\\Ctwo}_{\\bigstar}(\\Sigma^{-D_n\\rho\n -n-2(1-\\sigma)}\\Z_{(2)}^{BP\\R \\langle n \\rangle}). $$\n\\end{thm}\n\\vspace{0.3cm}\n\\begin{thm}\\label{thm:ER(n)}\nFor each $n\\geq 1$ the $\\Ctwo$-spectrum $E\\R(n)$ has Gorenstein duality of shift\n$-D_n\\rho -(n-1)-2(1-\\sigma)$. This means that\n\\begin{align*}\n \\Z_{(2)}^{E\\R(n)} &\\simeq \\Sigma^{D_n\\rho+(n-1)+2(1-\\sigma)} \\Gamma_{\\Jb_{n-1}}E\\R(n) \\\\\n &\\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3}\\Gamma_{J_{n-1}}E\\R(n) ,\n\\end{align*}\nfor $J_{n-1} = \\Jb_{n-1}\\cap\\pi_*^{C_2}E\\R(n)$. This induces likewise a local cohomology spectral sequence.\n\\end{thm}\n\nWe note that this has implications for the $\\Ctwo$-fixed point spectrum\n$(BP\\R \\langle n \\rangle)^{\\Ctwo}=BPR\\langle n\\rangle$. The graded ring\n$$\\pi_*(BPR\\langle n\\rangle)=\\pi_*^{\\Ctwo}(BP\\R \\langle n \\rangle)$$\nis the integer part of the $RO(\\Ctwo)$-graded coefficient ring\n$\\pi^{\\Ctwo}_{\\bigstar}(BP\\R \\langle n \\rangle)$. However, since the ideal\n$\\Jb_n$ is not generated in integer degrees, the statement for $BPR\\langle n\\rangle$\nis usually rather complicated, and one of our main messages is that\nworking with the equivariant spectra gives more insight. On the other hand, $ER(n) = E\\R(n)^{C_2}$ has integral Gorenstein duality because one can use the additional periodicity to move the representation suspension and the ideal $\\Jb_n$ to integral degrees. \n\nWe will discuss the general result in more detail later, but the two\nfirst cases are about familiar ring spectra. \n\n\\begin{example} (See Sections \\ref{sec:kR} and \\ref{sec:kRlcss}.) \nFor $n=1$, connective $K$-theory with Reality $k\\R$ is $2$-locally a\nform of $BP\\R\\langle 1\\rangle$.\n For this example we can work without\n2-localization, so that $\\Z$ means the integers. Our first theorem states\nthat $k\\R$ is Gorenstein of shift $-\\rho-1=-2-\\sigma$\nand that it has Gorenstein duality of shift $-4+\\sigma$. This just means that \n$$\\Z^{k\\R} \\simeq \\Sigma^{4-\\sigma} \\fib(k\\R \\to K\\R).$$\nThe local cohomology spectral sequence collapses to a short exact sequence associated to the fibre sequence just mentioned. We\nwill see in Section \\ref{sec:kRlcss} that the sequence is not split, even as abelian groups. \n\nTheorem \\ref{thm:ER(n)} recovers the main result of \\cite{H-S14},\ni.e.\\ that $\\Z^{K\\R} \\simeq \\Sigma^4K\\R$, which also implies $\\Z^{KO}\n\\simeq \\Sigma^4 KO$. It is a special feature of the case $n=1$ that we\nalso get a nice duality statement for the fixed points in the\nconnective case. Indeed, by considering the $RO(C_2)$-graded homotopy\ngroups of $k\\R$, one sees \\cite[3.4.2]{B-G10} that\n$$(k\\R \\otimes S^{-\\sigma})^{\\Ctwo}\\simeq \\Sigma^{1}ko. $$\nThis implies that connective $ko$ has untwisted Gorenstein duality of shift\n$-5$, i.e.\\ that \n$$\\Z^{ko} \\simeq \\Sigma^5\\fib(ko\\to KO).$$ \nThis admits a closely related non-equivariant proof, combining \nthe fact that $ku$ is Gorenstein (clear from coefficients) and the\nfact that complexification $ko\\longrightarrow ku$ is relatively Gorenstein\n(connective version of Wood's theorem \\cite[4.1.2]{B-G10}). \n\\end{example}\n\n\n\\begin{example}\n(See Examples \\ref{exa:forms} and \\ref{exa:tmf} or Lemma \\ref{lem:BPRnGordish} and Corollary \\ref{cor:BPRnGorDdish}.) \nThe 2-localization of the $C_2$-spectrum $tmf_1(3)$ is a form of $BP\\R\\langle\n2\\rangle$, and the theorem is closely related to results in\n\\cite{HM}. It states\nthat $tmf_1(3)$ is Gorenstein of shift $-4\\rho-2=-6-4\\sigma $\nand has Gorenstein duality of shift $-8-2\\sigma$. We show in Section\n\\ref{sec:tmfotlcss} that there are non-trivial \ndifferentials in the local cohomology spectral\nsequence.\n\nPassing to fixed points we obtain the 2-local equivalence\n$$BPR\\langle 2\\rangle=(BP\\R\\langle 2\\rangle)^{\\Ctwo}=tmf_0(3).$$\nBy contrast with the $n=1$ case, as observed in \\cite{HM}, $tmf_0(3)$ does not have untwisted\nGorenstein duality of any integer degree. \n\nA variant of Theorem \\ref{thm:ER(n)} also computes the\n$C_2$-equivariant Anderson dual of $TMF_1(3)$ and the computation of\nthe Anderson dual of $Tmf_1(3)$ from \\cite{HM} follows as well.\n\nThe results apply to $tmf_1(3)$ and $TMF_1(3)$ themselves (i.e.,\nwith just 3 inverted, and not all other odd primes). \n\\end{example}\n\nWe remark that our main theorem also recovers the main result of \\cite{Ricka} about the Anderson self-duality of integral Real Morava K-theory. \n\n\n\\subsection{Guide to the reader}\nWhile the basic structure of this paper is easily visible from the table of contents, we want to comment on a few features. \n\nThe precise statements of our main results can be found in Section \\ref{sec:results}. We will give two different proofs of them. One (Part 3) might be called `the hands on approach' which is\nelementary and explicit, and one (Part 2) uses Gorenstein techniques\ninspired by commutative algebra. \nThe intricacy and dependence \non specific calculations in the explicit approach and the make the conceptual approach\nvaluable. The subtlety of the structural requirements of the\nconceptual approach make the reassurance of the explicit approach\nvaluable. The exact results proved in Parts 2 and 3 are also\nslightly different. \n\nWhile the Gorenstein approach only relies on the knowledge of the homotopy groups of $H\\Zu$ and the reduction theorem Corollary \\ref{cor:reduction}, we need detailed information about the homotopy groups of quotients of $BP\\R$ for the hands-on approach. In Appendix \\ref{Appendix}, we give a streamlined account of the computation of $\\pi_{\\bigstar}^{C_2}BP\\R$ (which appeared first in \\cite{HK}). In Section \\ref{sec:BPRn}, we give a rather self-contained account of the homotopy groups of $BP\\R \\langle n \\rangle$ and of other quotients of $BP\\R$, which can also be read independently of the rest of the paper. While some of this is rather technical, most of the time we just have to use Corollary \\ref{Cor:crucial} whose statement (though not proof, perhaps) is easy to understand.\n\nWe give separate arguments for the computation of the Anderson dual of\n$k\\R$ so that this easier case might illustrate the more complicated\narguments of our more general theorems. Thus, if the reader is only\ninterested in $k\\R$, he or she might ignore most of this paper. More\nprecisely, under this assumption one might proceed as follows: First one looks at Section \\ref{sec:kRgroups} for a quick reminder on $\\pi_{\\bigstar}^{C_2}k\\R$, then one skims through Sections \\ref{sec:Basics} and \\ref{sec:AKG} to pick up the relevant definitions and then one proceeds directly to Section \\ref{sec:kR} or Section \\ref{sec:kRagain} to get the proof of the main result in the case of $k\\R$. Afterwards one may look at the pictures and computations in the rest of Section \\ref{sec:kRlcss} to see what happens behind the scenes of Gorenstein duality. \n\n\n\\vspace{1cm}\n\\part{Preliminaries and results}\\vspace{0.5cm}\n\\section{Basics and conventions about $C_2$-spectra}\\label{sec:Basics}\n\\subsection{Basics and conventions}\nWe will work in the homotopy category of genuine $G$-spectra (i.e., stable for suspensions by $S^V$ for any finite\ndimensional representation $V$) for $G =\\Ctwo$, the group of order $2$. We will denote by $\\otimes$ the derived smash product of spectra.\n\nWe may combine the equivariant and non-equivariant homotopy groups of\na $C_2$-spectrum into a Mackey functor, which we denote by $\\underline{\\pi}_*^{C_2}X$ and denote $C_2$-equivariant and underlying homotopy groups correspondingly by $\\pi^{C_2}_*X$ and $\\pi^e_*X$. For an abelian group $A$, we write $\\underline{A}$ for the constant Mackey functor (i.e., restriction maps are the identity),\nand $\\underline{A}^*$ for its dual (i.e., induction maps are the identity). \nWe write $HM$ for the Eilenberg-MacLane spectrum associated to a\nMackey functor $M$. \n\nAnother $C_2$-spectrum of interest to us is $k\\R$, the $C_2$-equivariant connective cover of\nAtiyah's $K$-theory with Reality \\cite{Ati66}. The term ``Real\nspectra'' derives from this example. The examples of Real bordism and the other $C_2$-spectra derived from it will be discussed in Section \\ref{sec:BPRn}.\n\nWe will usually grade our homotopy groups by the real\nrepresentation ring $RO(\\Ctwo)$, and we write $M_{\\bigstar}$ for $RO(\\Ctwo)$-graded groups. In addition to the real sign representation $\\sigma$ and the regular representation $\\rho$ the virtual\nrepresentation $\\pp =1-\\sigma$ is also significant. Examples of\n$RO(\\Ctwo)$-graded homotopy classes are the geometric Euler classes\n$a_V\\colon S^0 \\longrightarrow S^V$; in particular, $a=a_{\\sigma}$ will play a central role. In addition to $a$, we will also often have a class $u=u_{2\\sigma}$ of degree $2\\pp$. \n\nWe often want to be able to discuss gradings by certain subsets of\n$RO(\\Ctwo)$. To start with we often want to refer to gradings by multiples\nof the regular representation (where we write $M_{*\\rho}$), but we also\nneed to discuss gradings of the form $k\\rho -1$. For this, we use the notation\n$$*\\rho - =\\{ k\\rho \\; | \\; k\\in \\Z\\}\\cup \\{ k\\rho -1\\; | \\; k\\in \\Z\\}. $$\nFollowing \\cite{HM} we call an $RO(\\Ctwo)$-graded object $M$ {\\em even} if\n$M_{k\\rho -1}=0$ for all $k$. An $RO(\\Ctwo)$-graded Mackey functor is {\\em\n strongly even} if it is even and all the Mackey functors in gradings\n$k\\rho$ are constant. We call a $C_2$-spectrum (strongly) even if its homotopy groups are (strongly) even.\n\nIf $X$ is a strongly even $C_2$-spectrum and $x\\in \\pi_{2k}X$, we denote by $\\overline{x}$ its counterpart in $\\pi_{k\\rho}^{C_2}X$. If we want to stress that we consider a certain spectrum as a $C_2$-spectrum, we will also sometimes indicate this by a bar; for example, we may write $\\overline{tmf_1(3)}$ if we want to stress the $C_2$-structure of $tmf_1(3)$. \n\n\n\\subsection{Cellularity} \\label{sec:Cell}\n\\label{subsec:Rcell}\nIn a general triangulated category, it is conventional to say $M$ is \\emph{$K$-cellular} if\n$M$ is in the localizing subcategory generated by $K$ (or equivalently\nby all integer suspensions of $K$). A reference in the case of spectra is \\cite[Sec 4.1]{DGI}. We say that a $C_2$-spectrum $M$ is \\emph{$K$-$\\R$-cellular} (for a $C_2$-spectrum $K$)\nif it is in the localizing subcategory generated by the suspensions\n$S^{k\\rho}\\otimes K$ for all integers $k$. We note that this is the\nsame as the localizing subcategory generated by integer suspensions of\n$K$ and $(\\Ctwo)_+\\otimes K$ because of the cofibre sequence\n$$(C_2)_+ \\to S^0 \\to S^\\sigma.$$\nWe say that a map $N\\to M$ is a \\emph{$K$-$\\R$-cellularization} if $N$ is $K$-$\\R$-cellular and the induced map\n$$\\mathrm{Hom}(K, N) \\to \\mathrm{Hom}(K, M)$$\nis an equivalence of $C_2$-spectra. The $K$-$\\R$-cellularization is clearly unique up to equivalence. \n\nWe note that cellularity and $\\R$-cellularity are definitely\ndifferent. For example $(\\Ctwo)_+$ is not $S^0$-cellular, but it is\n$S^0$-$\\R$-cellular. \n\nIn this article, we will only use $\\R$-cellularity. \n\n\\subsection{The slice filtration}\n\\label{subsec:slice}\nRecall from \\cite[Section 4.1]{HHR} or \\cite{SlicePrimer} that the \\emph{slice cells} are the $\\Ctwo$-spectra of the form \n\\begin{itemize}\n \\item $S^{k\\rho}$ of dimension $2k$,\n \\item $S^{k\\rho-1}$ of dimension $2k-1$, and\n \\item $S^k\\otimes (\\Ctwo)_+$ of dimension $k$.\n\\end{itemize}\nA $\\Ctwo$-spectrum $X$ is $\\leq k$ if for every slice cell $W$ of dimension $\\geq k+1$ the mapping space $\\Omega^\\infty \\mathrm{Hom}_{\\mathbb{S}}(W, X)$ is equivariantly contractible. As explained in \\cite[Section 4.2]{HHR}, this leads to the definition of $X \\to P^kX$, which is the universal map into a $\\Ctwo$-spectrum that is $\\leq k$. The fibre of $$X \\to P^kX$$\nis denoted by $P_{k+1}X$. The $k$\\emph{-slice} $P_k^kX$ is defined as the fibre of $$P^kX\\to P^{k-1}X$$\nor, equivalently, as the cofibre of the map $P_{k+1}X \\to P_kX$. We have the following two useful propositions:\n\n\\begin{prop}[\\cite{SlicePrimer}, Cor 2.12, Thm 2.18]\n\\label{prop:sliceodd}\n If $X$ is an even $\\Ctwo$-spectrum, then $P^{2k-1}_{2k-1}X = 0$ for all $k\\in \\Z$.\n\\end{prop}\n\n\\begin{prop}[\\cite{SlicePrimer}, Cor 2.16, Thm 2.18]\n\\label{prop:sliceven}\n If $X$ is a $\\Ctwo$-spectrum such that the restriction map in\n $\\underline{\\pi}^{\\Ctwo}_{k\\rho}$ is injective, then $P^{2k}_{2k}X$ is\n equivalent to the Eilenberg-MacLane spectrum\n $\\underline{\\pi}^{\\Ctwo}_{k\\rho} X$.\n\\end{prop}\n\nThis allows us to give a characterization of an Eilenberg-MacLane spectrum based on regular representation degrees.\n\\begin{cor}\n\\label{cor:characterizingZ}\nAny even $\\Ctwo$-spectrum $X$ with \n$$\\underline{\\pi}^{\\Ctwo}_{k\\rho}(X)=\\begin{cases}\\underline{A} & \\text{ if } k= 0 \\\\\n0 & \\text{ else}\\end{cases}$$ \n for an abelian group $A$ is equivalent to $H\\underline{A}$.\n\\end{cor}\n\\begin{proof}\nBy the last two propositions, we have\n\\[P^k_k X\\simeq \\begin{cases} H\\underline{A} & \\text{ if } k=0 \\\\\n 0 & \\text{ else}\n \\end{cases}\n\\]\nBy the convergence of the slice spectral sequence \\cite[Theorem 4.42]{HHR}, the result follows.\n\\end{proof}\n\n\\section{Anderson duality, Koszul complexes and Gorenstein duality}\\label{sec:AKG}\n\n\\subsection{Duality for abelian groups}\\label{sec:DAb}\nIt is convenient to establish some conventions for abelian groups to\nstart with, so as to fix notation. \n\nPontrjagin duality is defined for all graded abelian groups $A$\nby \n$$A^{\\vee}=\\mathrm{Hom}_{\\Z}(A, \\Q \/\\Z). $$\nSimilarly, the rational dual is defined by \n$$A^{\\vee \\Q}=\\mathrm{Hom}_{\\Z}(A, \\Q ). $$\n\nSince $\\Q $ and $\\Q \/\\Z$ are injective abelian groups these two\ndualities are homotopy invariant and pass to the\nderived category. Finally the Anderson dual $A^*$ is defined by\napplying $\\mathrm{Hom}_{\\Z}(A, \\cdot )$ to the exact sequence\n$$0\\longrightarrow \\Z \\longrightarrow \\Q \\longrightarrow \\Q\/\\Z\\longrightarrow 0$$ \nso that we have a triangle \n$$A^*\\longrightarrow A^{\\vee \\Q }\\longrightarrow A^{\\vee }. $$\n\n\nIf $M$ is a free abelian group, then the\nAnderson dual is simply calculated by \n$$M^*=\\mathrm{Hom}_{\\Z}(M, \\Z)$$\n(since $M$ is free, the $\\mathrm{Hom}$ need not be derived). \n\nIf $M$ is a graded abelian group which is an $\\mathbb{F}_2$-vector\nspace then up to suspension the Anderson dual is the vector space dual: \n$$M^{\\vee}=\\mathrm{Hom}_{\\mathbb{F}_2}(M, \\mathbb{F}_2)\\simeq \\Sigma^{-1} M^*$$\n(since vector spaces are free, $\\mathrm{Hom}$ need not be derived). \n\n\n\\subsection{Anderson duality}\n\\label{subsec:Anderson}\nAnderson duality is the attempt to topologically realize the algebraic duality from the last subsection. It appears that it was invented by Anderson (only published in mimeographed notes \\cite{Anderson}) and Kainen \\cite{Kainen}, with similar ideas by Brown and Comenetz \\cite{B-C76}. For brevity and consistency, we will only use the term Anderson duality instead of Anderson--Kainen duality or Anderson--Brown--Comenetz duality throughout. We will work in the category of $\\Ctwo$-spectra, where Anderson duality was first explored by Ricka in \\cite{Ricka}. \n\nLet $I$ be an injective abelian group. Then we let $I^{\\mathbb{S}}$ denote the $C_2$-spectrum representing the functor \n$$X \\mapsto \\mathrm{Hom}(\\pi_{*}^{C_2}X,I).$$\nFor an arbitrary $C_2$-spectrum, we define $I^X$ as the function spectrum $F(X, I^{\\mathbb{S}})$. For a general abelian group $A$, we choose an injective resolution \n$$A \\to I \\to J$$\nand define $A^X$ as the fibre of the map $I^X \\to J^X$. For example, we get a fibre sequence\n$$\\Z^X\\longrightarrow \\Q^X\\longrightarrow (\\Q\/\\Z)^X.$$\nIn general, we get a short exact sequence of homotopy groups\n$$0 \\to \\mathrm{Ext}_{\\Z}(\\pi_{-k-1}^{C_2}(X), A) \\to \\pi_k^{C_2} (A^X) \\to \\mathrm{Hom}(\\pi_{-k}^{C_2}(X), A) \\to 0.$$\nThe analogous exact sequence is true for $RO(C_2)$-graded Mackey functor valued homotopy groups by replacing $X$ by $(C_2\/H)_+ \\wedge \\Sigma^VX$. \nOur most common choices will be $A = \\Z$ and $A=\\Z_{(2)}$.\n\nFrom time to time we we use the following property of Anderson duality: If $R$ is a strictly commutative $C_2$-ring spectrum and $M$ an $R$-module, then $\\mathrm{Hom}_R(M, A^R) \\simeq A^M$ as $R$-modules as can easily be seen by adjunction.\n\nOne of the reasons to consider Anderson duality is that it provides universal coefficient sequences. In the $C_2$-equivariant world, this takes the following form \\cite[Proposition 3.11]{Ricka}:\n\\[0 \\to \\mathrm{Ext}_\\Z^1(E_{\\alpha-1}^{\\Ctwo}(X), A) \\to (A^E)_{\\alpha}^{\\Ctwo}(X) \\to \\mathrm{Hom}_\\Z(E_{\\alpha}^{\\Ctwo}(X), A) \\to 0,\\]\nwhere $E$ and $X$ are $C_2$-spectra, $\\alpha\\in RO(C_2)$ and $A$ is an abelian group. \n\nOur first computation is the Anderson dual of the Eilenberg--MacLane\nspectrum of the constant Mackey functor $\\underline{\\Z}$.\n\n\\begin{lemma}\n\\label{lem:Zu}\nThe Anderson dual of the Eilenberg-MacLane spectrum $H\\Zu$ (as an\n$\\bbS$-module) is given by \n$$\\Z^{H\\Zu}\\simeqH\\Zu^* \\simeq \\Sigma^{2\\pp} H\\Zu, $$\nwhere $\\delta = 1-\\sigma$.\n\\end{lemma}\n\n\\begin{proof}\nThe first equivalence follows from the isomorphisms\n$$\\underline{\\pi}^{C_2}_*(\\Z^{H\\Zu}) \\cong \\mathrm{Hom}_{\\Z}(\\underline{\\pi}^{C_2}_{-*}H\\Zu, \\Z) \\cong \\underline{\\Z}^*.$$\n\nSince \n$$\\pi_*^{\\Ctwo}(S^{2-2\\sigma}\\otimes H\\Zu)=H^*_{\\Ctwo}(S^{2\\sigma -2};\n\\underline{\\Z})=H^*(S^{2\\sigma -2}\/\\Ctwo; \\Z), $$\nand $S^{2\\sigma}=S^0*S(2\\sigma)$ is the unreduced suspension of of $S(2\\sigma)$, the second equivalence is a calculation of\nthe cohomology of $\\R P^1$ . \n\\end{proof}\n\n\n\\begin{remark}\nThis proof shows that if $\\Ctwo$ is replaced by a cyclic group of any order\n we still have\n$$\\Z^{H\\Zu}=H\\Zu^* \\simeq \\Sigma^{\\lambda} H\\Zu$$\nwhere $\\lambda =\\eps -\\alpha$ (with $\\eps$ the trivial one\ndimensional complex representation and $\\alpha$ a faithful one\ndimensional representation). \n\\end{remark}\n\nAnderson duality works, of course, also for non-equivariant spectra. We learned the following proposition comparing the equivariant and non-equivariant version in a conversation with Nicolas Ricka.\n\n\\begin{prop}\\label{prop:AndersonFixed}Let $A$ be an abelian group. We have $(A^X)^{C_2} \\simeq A^{(X^{C_2})}$ for every $C_2$-spectrum $X$. \n\\end{prop}\n\\begin{proof}Let $\\infl_e^{C_2} Y$ denote the inflation of a spectrum $Y$ to a $C_2$-spectrum with `trivial action', i.e.\\ the left derived functor of first regarding it as a naive $C_2$-spectrum with trivial action and then changing the universe. This is the (derived) left adjoint for the fixed point functor \\cite[Prop 3.4]{MM02}.\n\nLet $I$ be an injective abelian group. Then there is for every spectrum $Y$ a natural isomorphism \n\\begin{align*}\n[Y, (I^X)^{C_2}] &\\cong [\\infl_e^{C_2} Y, I^X]^{C_2} \\\\\n&\\cong \\mathrm{Hom}(\\pi_0^{C_2}(\\infl_e^{C_2} Y \\otimes X), I)\\\\\n&\\cong \\mathrm{Hom}(\\pi_0(Y\\otimes X^{C_2}), I) \\\\\n&\\cong [Y, I^{(X^{C_2})}].\n\\end{align*}\nHere, we use that fixed points commute with filtered homotopy colimits and cofibre sequences and therefore also with smashing with a spectrum with trivial action. Thus, there is a canonical isomorphism in the homotopy category of spectra between $I^{(X^{C_2})}$ and $(I^X)^{C_2}$ that is also functorial in $I$ (by Yoneda). For a general abelian group $A$, we can write $A^{(X^{C_2})}$ as the fibre of $(I^0)^{X^{C_2}} \\to (I^1)^{X^{C_2}}$ (and similarly for the other side) for an injective resolution $0\\to A\\to I^0\\to I^1$. Thus, we obtain a (possibly non-canonical) equivalence between $A^{(X^{C_2})}$ and $(A^X)^{C_2}$.\n\\end{proof}\n\\begin{remark}An analogous result holds, of course, for every finite group $G$. \n\\end{remark} \n\n\\subsection{Koszul complexes and derived power torsion}\\label{sec:Koszul}\nLet $R$ be a non-equivariantly $E_{\\infty}$ $C_2$-ring spectrum and $M$ be an\n$R$-module. In this section we will recall two versions of stable\nKoszul complexes. Among their merits is that they provide models for\ncellularization or $\\R$-cellularization in cases of interest for us. A basic reference for the material in this section is \\cite{G-M95}. \n\nAs classically, the $r$-power torsion in a module $N$ can be defined as the kernel of $N \\to N[\\frac1r]$, we define the \\emph{derived $J$-power torsion} of $M$ with respect to an ideal $J = (x_1,\\dots, x_n) \\subseteq \\pi^{\\Ctwo}_{\\bigstar}(R)$ as\n\\begin{align*}\\Gamma_J M = \\fib(R \\to R[\\frac1{x_1}]) \\otimes_R \\cdots\n \\otimes_R \\fib(R \\to R[\\frac1{x_n}]) \\otimes_R M .\n\\end{align*}\nThis is also sometimes called the \\emph{stable Koszul complex} and also denoted by $K(x_1,\\dots, x_n)$. As shown in \\cite[Section 3]{G-M95}, this only depends on the ideal $J$ and not on the chosen generators. As algebraically, the derived functors of $J$-power torsion are the local cohomology groups, we might expect a spectral sequence computing the homotopy groups of $\\Gamma_J M$ in terms of local cohomology. As in \\cite[Section 3]{G-M95}, this takes the form\n\\begin{align}\\label{eqn:localcohomology} H_J^s(\\pi_{\\bigstar+V}^{C_2}M) \\Rightarrow \\pi_{V-s}^{C_2}(\\Gamma_JM).\\end{align}\n\nOur second version of the Koszul complex can be defined in the\none-generator case as\n$$\\kappa_R(x) = \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{(1-l)|x|}R\/x^l$$\nfor $x \\in \\pi^{\\Ctwo}_{\\bigstar}(R)$.\nHere, the map $R\/x^l \\to \\Sigma^{-|x|}R\/x^{l+1}$ is induced by the diagram of cofibre sequences\n\\[\n \\xymatrix{\\Sigma^{|x^l|}R \\ar[r]^{x^l}\\ar[d]^=& R \\ar[r]\\ar[d]^x & R\/x^l\\ar@{-->}[d] \\\\\n \\Sigma^{|x^l|}R \\ar[r]^{x^{l+1}} & \\Sigma^{-|x|}R \\ar[r] & \\Sigma^{-|x|}R\/x^{l+1}\n }\n\\]\n\nMore generally, we can make for a sequence $\\mathbf{x} = (x_1,\\dots, x_n)$ in $\\pi^{\\Ctwo}_{\\bigstar}(R)$ the definition\n\\begin{align*}\\kappa_R(\\mathbf{x}; M) &:= \\kappa_R(x_1)\\otimes_R\\cdots \\otimes_R \\kappa_R(x_n)\\otimes_R M\\\\\n&\\simeq \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{-((l_1-1)+\\cdots (l_n-1))|x|} M\/(x_1^{l_1},\\dots, x_n^{l_n})\\end{align*}\n\nThe spectrum $\\kappa_R(x)$ comes with an obvious filtration by $\\Sigma^{(1-l)|x|}R\/x^l$ with filtration quotients $\\Sigma^{-l|x|}R\/x$. We can smash these filtrations together to obtain a filtration of $\\kappa_R(\\mathbf{x})$\nwith filtration quotients wedges of summands of the form $\\Sigma^{-l_1|x_1| -\\cdots - l_n|x_n|}R\/(x_1,\\dots, x_n)$ (see \\cite[1.3.11-12]{tilson} or \\cite[2.8, 2.12]{tilsonArXiv}). Using the following lemma, we obtain also a corresponding filtration on $\\Gamma_JR$. \n\n\\begin{lemma}\\label{lem:Koszul} For $\\mathbf{x}$ as above, we have \n\\begin{align*}\n \\kappa_R(\\mathbf{x}) &\\simeq \\Sigma^{|x_1|+\\cdots +|x_n|+n}\\Gamma_JR.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 3.6]{G-M95}.\n\\end{proof}\n\nWe can also define $\\kappa_R(\\mathbf{x};M)$ (and likewise the other versions of Koszul complexes) for an infinite sequence of $x_i$ by just taking the filtered homotopy colimit over all finite subsequences. Usually Lemma \\ref{lem:Koszul} breaks down in the infinite case.\n\n\\begin{remark}\\label{rmk:hocolim}\n The homotopy colimit defining $\\kappa_R(\\mathbf{x};M)$ has a directed cofinal subsystem, both in the\nfinite and in the infinite case. Indeed, the colimit ranges over all sequences $(l_1,l_2,\\dots)$ with only finitely many entries nonzero. For the directed subsystem, we start with $(0,0,\\dots)$ and raise in the $n$-th step the first $n$ entries by $1$. Directed homotopy colimit are well-known to be weak colimits in the homotopy category of $R$-modules, i.e.\\ every system of compatible maps induces a (possibly non-unique) map from the homotopy colimit \\cite[Sec 3.1]{Mar83} \\cite[Sec II.5]{Sch07}.\n\\end{remark}\n\nOne of the reason for introducing $\\Gamma_JM$ is that it provides a model for the $\\R$-cellularization of $M$ with respect to $R\/J = (R\/x_1) \\otimes_R \\cdots \\otimes_R (R\/x_n)$ in the sense of Section \\ref{sec:Cell}.\n\\begin{prop}\\label{prop:cell}\n Suppose that $x_1\\dots, x_n \\in \\pi_{*\\rho}^{C_2}R$ and set $J = (x_1,\\dots, x_n)$. Then $\\Gamma_JM \\to M$ is a $\\R$-cellularization with respect to $R\/J$ in the (triangulated) category of $R$-modules. \n\\end{prop}\n\\begin{proof}\n Clearly, $\\kappa_R(x_1,\\dots, x_n; M)$ is $\\R$-cellular with respect to $M\/J$; furthermore $M\/J$ is $R\/J$-$\\R$-cellular as clearly $M$ is $R$-cellular. To finish the proof, we have to show that \n $$\\mathrm{Hom}_R(R\/J, \\Gamma_JM) \\to \\mathrm{Hom}_R(R\/J, M)$$\n is an equivalence. Note that $\\Gamma_JM = \\Gamma_{x_n}(\\Gamma_{(x_1,\\dots, x_{n-1})}M)$. Thus, it suffices by induction to show that \n $$\\mathrm{Hom}_R(A\/x, \\Gamma_x B) \\to \\mathrm{Hom}_R(A\/x, B)$$\n is an equivalence for all $R$-modules $A,B$. This is equivalent to \n $$\\mathrm{Hom}_R(A\/x, B[x^{-1}]) = 0$$\nwhich is true as multiplication by $x$ induces an equivalence\n $$\\mathrm{Hom}_R(A,B[x^{-1}]) \\xrightarrow{x^*} \\mathrm{Hom}_R(\\Sigma^{|x|}A, B[x^{-1}]).\\qedhere$$\n\\end{proof}\n\n\\begin{cor}\\label{cor:cellular}\n Let $M$ be a connective $R$-module and $A$ an abelian group. Then the Anderson dual $A^M$ is $\\R$-cellular with respect to $R\/J$ for every ideal $J\\subset \\pi_{\\bigstar}^{C_2}$ finitely generated in degrees $a+b\\sigma$ with $a\\geq 1$ and $a+b\\geq 1$. \n\\end{cor}\n\\begin{proof}\n By the last proposition, we have to show that $\\Gamma_JA^M \\simeq A^M$. For this it suffices to show that $A^M[x^{-1}]$ is contractible for every generator $x$ of $J$. As $M$ is connective, we know that $\\pi_{a+b\\sigma}M = 0$ if $a<0$ and $a+b<0$ (this follows, for example, using the cofibre sequence $(C_2)_+ \\to S^0 \\to S^\\sigma$). Thus, $\\pi_{a+b\\sigma}A^M = 0$ if $a>0$ and $a+b>0$. The result follows. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Real bordism and the spectra $BP\\mathbb{R}\\langle n \\rangle$}\\label{sec:BPRn}\n\\subsection{Basics and homotopy fixed points}\\label{sec:BPRBasics}\nThe $\\Ctwo$-spectrum $M\\R$ of \\emph{Real bordism} was originally defined by Araki and Landweber. In the naive model of $\\Ctwo$-spectra, \nwhere a $\\Ctwo$-spectrum is just given as a sequence $(X_n)$\n of pointed $\\Ctwo$-spaces together with maps \n $$\\Sigma^{\\rho}X_n \\to X_{n+1}$$\n it is just given by the Thom spaces $M\\R_n = BU(n)^{\\gamma_n}$ \n with complex conjugation as $\\Ctwo$-action. Defining it as a strictly commutative $\\Ctwo$-orthogonal spectrum requires more care and was done \n in \\cite[Example 2.14]{SchEquiv} and \\cite[Section B.12]{HHR}. An important fact is that the geometric fixed points of $M\\R$ are equivalent to \n $MO$ (first proven in \\cite{A-M} and reproven in \\cite[Proposition B.253]{HHR}).\n \nAs shown in \\cite{Ara79} and \\cite[Theorem 2.33]{HK}, there is a\nsplitting \n$$M\\R_{(2)} \\simeq \\bigoplus_{i}\\Sigma^{m_i\\rho}BP\\R,$$\nwhere the underlying spectrum of $BP\\R$ agrees with $BP$. This\nsplitting corresponds on geometric fixed points to the splitting \n$$MO \\simeq \\bigoplus_{i}\\Sigma^{m_i}H\\F_2.$$ \nAs shown in \\cite{HK} (see also\nAppendix \\ref{Appendix}), the restriction map \n$$\\pi_{*\\rho}^{\\Ctwo}BP\\R\\to \\pi_{2*}BP$$ \nis an isomorphism. Choose now arbitrary\nindecomposables $v_i \\in \\pi_{2(2^i-1)}BP$ and denote their lifts to\n$\\pi_{(2^i-1)\\rho}^{\\Ctwo}BP\\R$ and their images in $\\pi_{(2^i-1)\\rho}^{\\Ctwo}M\\R$\nby $\\vb_i$. We denote by $BP\\R \\langle n \\rangle$ the quotient \n$$BP\\R\/(\\vb_{n+1},\\vb_{n+2},\\dots)$$ \nin the homotopy category of $M\\R$-modules. At least a priori, this depends on the choice of $v_i$. \n\nWe want to understand the homotopy groups of $BP\\R \\langle n \\rangle$. This was first\ndone by Hu in \\cite{Hu} (beware though that Theorem 2.2 is not correct\nas stated there) and partially redone in \\cite{K-W13}. For the\nconvenience of the reader, we will give the computation again. Note\nthat our proofs are similar but not identical to the ones in the\nliterature. The main difference is that we do not use ascending\ninduction and prior knowledge of $H\\Z$ to compute $\\Phi^{C_2}BP\\R \\langle n \\rangle$,\nbut precise knowledge about $\\pi_{\\bigstar}^{C_2}BP\\R$ -- this is not\nsimpler than the original approach, but gives extra information about\nother quotients of $BP\\R$, which we will need later. We recommend that\nthe reader looks at Appendix A for a complete understanding of the\nresults that follow. \n\nWe will use the\nTate square \\cite{GMTate} and consider the following diagram in which\nthe rows are cofibre sequences: \n\n \\[\\xymatrix@C=0.7cm{\n BP\\R \\langle n \\rangle \\otimes (EC_2)_+ \\ar[r]\\ar[d]^\\simeq & BP\\R \\langle n \\rangle \\ar[r]\\ar[d]& BP\\R \\langle n \\rangle\\otimes \\tilde{E}C_2 \\ar[d]\\ar[r] &\\SigmaBP\\R \\langle n \\rangle\\otimes (EC_2)_+ \\ar[d]\\\\\n BP\\R \\langle n \\rangle^{(EC_2)_+} \\otimes (EC_2)_+ \\ar[r] & BP\\R \\langle n \\rangle^{(EC_2)_+} \\ar[r]& BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2 \\ar[r] & \\SigmaBP\\R \\langle n \\rangle\\otimes (EC_2)_+ \n }\n \\]\n\nAfter taking fixed points this becomes\n \\[\\xymatrix{\n BP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[r]\\ar[d]^= & BP\\R \\langle n \\rangle^{\\Ctwo} \\ar[r]\\ar[d]& BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\ar[d]\\ar[r] &\\SigmaBP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[d]\\\\\n BP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[r] & BP\\R \\langle n \\rangle^{h\\Ctwo} \\ar[r]& BP\\R \\langle n \\rangle^{t\\Ctwo} \\ar[r] & \\SigmaBP\\R \\langle n \\rangle_{h\\Ctwo} \n }\n \\]\n\n First, we compute the homotopy groups of the homotopy fixed points. For this we use the $RO(C_2)$-graded homotopy fixed point spectral sequence, described for example in \\cite[Section 2.3]{HM}.\n \n \\begin{prop}\\label{prop:BPRnHomotopy}\n The $RO(\\Ctwo)$-graded homotopy fixed point spectral sequence\n \\[E_2=H^*(\\Ctwo; \\pi^e_{\\bigstar}BP\\R \\langle n \\rangle) \\cong \\Z_{(2)}[\\vb_1,\\dots, \\vb_n, u^{\\pm 1}, a]\/2a \\Rightarrow \\pi^{\\Ctwo}_\\bigstar (BP\\R \\langle n \\rangle^{(E\\Ctwo)_+})\\]\n has differentials generated by $d_{2^{i+1}-1}(u^{2^{i-1}}) = a^{2^{i+1}-1}\\vb_i$ for $i\\leq n$ and $E_{2^{n+1}} = E_\\infty$.\n \\end{prop}\n \\begin{proof}\n The description of $E_{2^{n+1}}$ is entirely analogous to the proof of \\ref{prop:Differentials}, using that $a^{2^{i+1}-1}\\vb_i = 0$ in $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$. Now we need to show that there are no further differentials: As every element in filtration $f$ is divisible by $a^f$ in $E^{2^{n+1}}$, the existence of a nonzero $d_m$ (with $m\\geq 2^{n+1}$) implies the existence of a nonzero $d_m$ with source in the $0$-line. Moreover, a nonzero $d_m$ of some element $u^{l}\\vb$ (for $\\vb$ a polynomial in the $\\vb_i$) on the $0$-line implies a nonzero $d_m$ on $u^{l}$ as $\\vb$ is a permanent cycle (in the image from $BP\\R$). The image of such a differential must be of the form $a^mu^{l'}\\vb'$, where $\\vb'$ is a polynomial in $\\vb_1,\\dots, \\vb_n$. As $a^m\\vb_i = 0$ for $1\\leq i\\leq n$ in $E^{2^{n+1}}$, the polynomial $\\vb'$ must be constant. Counting degrees, we see that \n \\[(2l-1)-2l\\sigma = |u^l|-1 = |a^mu^{l'}| = 2l' -(2l'+m)\\sigma\\]\n and thus $m = 2l-2l' = 1$. This is clearly a contradiction. \n \\end{proof}\n \n \\begin{cor}\n We have \n $$\\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2) \\cong \\F_2[u^{\\pm 2^n}, a^{\\pm 1}].$$\n In particular, we get $\\pi_*BP\\R \\langle n \\rangle^{tC_2} \\cong \\F_2[x^{\\pm 1}]$, where $x = u^{2^n}a^{-2^{n+1}}$ and $|x| = 2^{n+1}$. These are understood to be additive isomorphisms.\n \\end{cor}\n \\begin{proof}\n Recall that \n $$\\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2) = \\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+})[a^{-1}].$$\n as $S^{\\infty\\sigma}$ is a model of $\\tilde{E}C_2$. The result follows as all $\\vb_i$ are $a$-power torsion, but $u^{2^nm}$ is not. \n \\end{proof}\n \n \\subsection{The homotopy groups of $BP\\mathbb{R}\\langle n\\rangle$}\\label{sec:BPRnC2}\n \n Computing the homotopy groups of the fixed points is more delicate\n than the computation of the homotopy fixed points. We first have to\n use our detailed knowledge about the homotopy groups of $BP\\R$.\n Given a sequence $\\underline{l} = (l_1,\\dots)$, we denote by\n $BP\\R\/\\underline{\\vb}^{\\underline{l}}$ the spectrum\n $BP\\R\/(\\vb^{l_{i_1}}_{i_1}, \\vb^{l_{i_2}}_{i_2},\\dots)$, where $i_j$\n runs over all indices such that $l_{i_j}\\neq 0$. Similarly\n $BP\\R\/\\vb_i^j$ is understood to be $BP\\R$ if $j=0$. We use the\n analogous convention when we have algebraic quotients of homotopy\n groups. \n\nWe recommend the reader skips the proof of the following result for\nfirst reading, as the technical detail is not particularly\nilluminating. \n \n \\begin{prop}\\label{prop:BPBound}\nLet $k\\geq 1$ and $\\underline{l} = (l_1,l_2, \\dots)$ be a sequence of nonnegative integers with $l_k=0$. Then the element $\\vb_k$ acts injectively on $(\\pi_{*\\rho-c}^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}}$ if $0\\leq c \\leq 2^{k+1}+1$, with a splitting on the level of $\\Z_{(2)}$-modules. \n\\end{prop}\n\\begin{proof}\nRecall from Appendix \\ref{Appendix} that $\\pi_\\bigstar^{\\Ctwo}BP\\R$ is isomorphic to the subalgebra of\n $$P\/(2a, \\vb_ia^{2^{i+1}-1})$$\n (where $i$ runs over all positive integers) generated by $\\vb_i(j) = u^{2^ij}\\vb_i$ (with $i,j\\in\\Z$ and $i\\geq 0$) and $a$, where $P = \\Z_{(2)}[a, \\vb_i, u^{\\pm 1}]$. The degrees of the elements are $|a| = 1-\\rho$ and $$|\\vb_i(j)| = (2^i-1)\\rho + 2^ij(4-2\\rho) = (2^i-1-2^{i+1}j)\\rho + 2^{i+2}j.$$\n We add the relations $\\vb_i^{l_i} = 0$ if $l_i\\neq 0$.\n \nWe will first show that the ideal of $\\vb_k$-torsion elements in\n$(\\pi_\\bigstar^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}}$ is\ncontained in the ideal generated by $a^{2^{k+1}-1}$ and\n$\\vb_s^{l_s-1}\\vb_s(j)$ for $s$ with $l_s \\neq 0$ and $j$ divisible by\n$2^{k-s}$ if $sk$. \n\nNow assume that $x$ is a $\\vb_k$-torsion element not divisible by $a^n$ for $n\\geq 2^{k+1}-1$. Then $x$ must be of the form $\\vb_s^{l_s-1}\\vb_s(j)<$ where $j$ is divisible by $2^{k-s}$ if $sk$, we see that if $|x|$ is of the form $*\\rho - c$ with $c\\geq 0$, then we have $$c \\geq 2^{k+2}-(2^{k+1}-2) = 2^{k+1}+2.$$ The statement about injectivity follows also in this case. \n\nWe still have to show the split injectivity. \nIf $\\vb_k y = 2z$, but $y$ not divisible by $2$, then $y$ must be of the form $2\\vb u^{2^kj}$ in $P$, where $\\vb$ is a polynomial in the $\\vb_i$. Thus, $|y| = 2^{k+2}j +*\\rho$, so we are fine in degree $*\\rho -c$ for $0\\leq c\\leq 2^{k+1}+1 \\leq 2^{k+2}-1$. \n\\end{proof}\n\n\\begin{remark}\n The exact bounds in the preceding proposition are not very important. The only important part for later inductive arguments is that the bound grows with $k$. \n\\end{remark}\n\n\\begin{cor}\\label{Cor:QuotientBP}\nLet $\\underline{l} = (l_1,l_2, \\dots)$ be a sequence with only finitely many nonzero entries and let $j$ be the smallest index such that $l_j \\neq 0$. Then the map \n\\[\n(\\pi_{*\\rho-c}^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}} \\to \\pi_{*\\rho-c}^{\\Ctwo}(BP\\R\/\\underline{\\vb}^{\\underline{l}})\n\\] \nis an isomorphism for $0\\leq c\\leq 2^{j+1}$.\n\\end{cor}\n\\begin{proof}\nWe use induction on the number $n$ of nonzero indices in $\\underline{l}$. If $n=0$ (and $j=\\infty$), the statement is clear.\n\nFor the step, define $\\underline{l}'$ to be the sequence obtained from $\\underline{l}$ by setting $l_j = 0$. Consider the short exact sequence\n\\[0 \\to (\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'}))\/\\vb_j^{l_j} \\to \\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}}) \\to \\left\\{\\pi_{*\\rho-(c+1)}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\right\\}_{\\vb_j^{l_j}} \\to 0.\\]\nHere, the notion $\\{X\\}_z$ denotes the elements in $X$ killed by $z$. \n\nAssume $c\\leq 2^{j+1}$. By the induction hypothesis, $\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\cong (\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}'}$ as $c\\leq 2^{j+2}$, so that $(\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'}))\/\\vb_j^{l_j} \\cong (\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}}$. Furthermore,\n\\begin{align*}\n\\left\\{\\pi_{*\\rho-(c+1)}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\right\\}_{\\vb_j^{l_j}} &\\cong \\left\\{(\\pi_{*\\rho-(c+1)}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}'}\\right\\}_{\\vb_j^{l_j}} \\\\\n&\\cong 0\n\\end{align*}\nas follows from $c+1 \\leq 2^{j+2}$ and $c+1 \\leq 2^{j+1}+1$ by the induction hypothesis and Proposition \\ref{prop:BPBound}. Thus, we see that $(\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}} \\to \\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}})$ is an isomorphism.\n\\end{proof}\n\nThe following corollary is crucial:\n\n\\begin{cor}\\label{Cor:crucial}\n Let $I \\subset \\Z_{(2)}[\\vb_1,\\dots]$ be an ideal generated by powers of the $\\vb_i$. Then $BP\\R\/I$ is strongly even. \n\\end{cor}\n\\begin{proof}\n As being strongly even is a property closed under filtered homotopy colimits, we are reduced to the case that $I$ is finitely generated. By the last corollary, it suffices to show that $BP\\R$ itself is strongly even. That the Mackey functor $\\underline{\\pi}_{*\\rho}^{\\Ctwo}(BP\\R)$ is constant is clear from Theorem \\ref{thm:BPR}. \n \n Assume that $x$ is a nonzero element in $\\pi_{*\\rho-1}^{\\Ctwo}BP\\R$. We can assume that $x$ is represented by $a^ku^l\\vb$ in the $E_2$-term of the homotopy fixed point spectral sequence for $BP\\R$, where $\\vb$ is a monomial in the $\\vb_i$ (with $\\vb_0 =2$), $k\\geq 0$ and $l\\in\\Z$. The element $x$ is in degree $k+4l + *\\rho$. Let $j\\geq 0$ be the minimal number such that $\\vb_j|\\vb$. Then $2^j|l$ and $k\\leq 2^{j+1}-2$. This is clearly in contradiction with $k+4l = -1$. \n\\end{proof}\n\nWe recover the $C_2$-case of the reduction theorem of \\cite[Prop 4.9]{HK} and \\cite[Thm 6.5]{HHR}.\n\\begin{cor}\\label{cor:reduction}\n There is an equivalence $BP\\R\/(\\vb_1,\\vb_2,\\dots) \\simeq H\\underline{\\Z}_{(2)}$.\n\\end{cor}\n\\begin{proof}\n This follows directly from the last corollary and Corollary \\ref{cor:characterizingZ}. \n\\end{proof}\n\n\n\n\\begin{cor}\\label{cor:rho+}\n Let $I \\subset \\Z_{(2)}[\\vb_1,\\dots]$ be an ideal generated by powers of the $\\vb_i$. Then \n $$\\pi_{*\\rho+1}^{C_2}BP\\R\/I \\cong \\F_2\\{a\\}\\otimes \\Z_{(2)}[\\vb_1, \\vb_2,\n \\dots]\/I.$$\n \\end{cor}\n \\begin{proof}\n As $BP\\R\/I$ is strongly even, this follows from \\cite[Lemma 2.15]{HM}.\n \\end{proof}\n\n\nThis allows us to compute $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle$.\n \n \\begin{prop}\\label{prop:BPRnFixedPoints}\n The spectrum $BP\\R \\langle n \\rangle$ is the connective cover of its Borel completion $BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$. The cofibre $C$ of $BP\\R \\langle n \\rangle \\to BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$ has homotopy groups $$\\pi_\\bigstar^{\\Ctwo}C \\cong \\F_2[a^{\\pm 1}, u^{-2^n}]u^{-2^n},$$\n with the naming of the elements indicating the map $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle^{(E\\Ctwo)_+} \\to \\pi_\\bigstar^{\\Ctwo}C$.\n \\end{prop}\n \\begin{proof}\n This is clear on underlying homotopy groups. Thus, we have only to show that $BP\\R \\langle n \\rangle^{\\Ctwo} \\to BP\\R \\langle n \\rangle^{h\\Ctwo}$ is a connective cover. For that purpose, it is enough to show that $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo}$ is connective and that the fibre of $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\to BP\\R \\langle n \\rangle^{t\\Ctwo}$ has homotopy groups only in negative degrees. \n \n We have $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\simeq BP\\R^{\\Phi \\Ctwo}\/(\\vb_{n+1},\\dots)$. As noted in the proof of Proposition \\ref{prop:vbn}, the image of $\\vb_i$ in $M\\R^{\\phi \\Ctwo}$ is $0$. As the quotient $BP\\R^{\\Phi \\Ctwo}\/\\vb_{n+1},\\dots$ can be taken in the category of $M\\R^{\\Phi \\Ctwo}$-modules, we are only quotiening out by $0$. It follows easily that $(BP\\R\/(\\vb_{n+1},\\dots, \\vb_{n+m}))^{\\Phi \\Ctwo}$ has in the homotopy groups an $\\F_2$ in all degrees of the form $\\sum_{i=n+1}^{n+m}\\varepsilon_i(|v_i|+1) = \\sum_{i=n+1}^{n+m}\\varepsilon_i2^i$ with $\\varepsilon_i = 0$ or $1$. As geometric fixed point commute with homotopy colimits, we see that $\\pi_*BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\cong \\F_2[y]$ (additively) with $|y| = 2^{n+1}$. It remains to show that $y^k$ maps nonzero to $\\pi_*BP\\R \\langle n \\rangle^{t\\Ctwo}$ (and hence maps to $x^k)$. \n \n It is enough to show that $a^{-|y^k|-1}y^k$ maps nonzero to $\\pi_\\bigstar^{\\Ctwo} \\Sigma BP\\R \\langle n \\rangle\\otimes (E\\Ctwo)_+$ in the sequence coming from the Tate square, i.e.\\ that $a^{-|y^k|-1}y^k$ is not in the image from (the fixed points of) $BP\\R \\langle n \\rangle$. But $a^{-|y^k|-1}y^k$ is in degree $(|y^k|+1)\\rho-1$ and $\\pi_{(|y^k|+1)\\rho-1}^{\\Ctwo}BP\\R \\langle n \\rangle = 0$ by Corollary \\ref{Cor:crucial}. \n \\end{proof}\n \n Let us describe the homotopy groups of $BP\\R \\langle n \\rangle$ in more detail. We set $\\vb_0 = 2$ for convenience. Denote by $BB$ (for \\emph{basic block}) the $\\Z_{(2)}[a, \\vb_1,\\dots,\\vb_n]\/2a$-submodule of \n $$\\Z_{(2)}[\\vb_1,\\dots, \\vb_n]\/(a^{2^{k+1}-1}\\vb_k)_{0\\leq k \\leq n}$$\n generated by $1$ and by $\\vb_k(m) = u^{2^km}\\vb_k$ for $0\\leq k < n$ and $0< m <2^{n-k}$.\nBy Proposition \\ref{prop:BPRnHomotopy}, we see that \n $$\\pi_{\\bigstar}^{C_2}BP\\R \\langle n \\rangle^{(EC_2)_+} \\cong BB[U^{\\pm 1}]$$\n with $U = u^{2^n}.$ Note that this isomorphism is not claimed to be multiplicative; in general, $BP\\R \\langle n \\rangle$ is not even known to have any kind of (homotopy unital) multiplication.\n \n Define $BB'$ to be the kernel of the map $BB \\to \\F_2[a]$ given by sending all $\\vb_k$ and $\\vb_k(m)$ to zero. Set $NB = \\Sigma^{\\sigma-1}\\F_2[a]^\\vee \\oplus BB'$, where $NB$ stands for \\emph{negative block}. Then it is easy to see from the last proposition that \n $$\\pi_{\\bigstar}^{C_2}BP\\R \\langle n \\rangle \\cong BB[U] \\oplus U^{-1}NB[U^{-1}],$$ \n where this isomorphism is again only meant additively. We will be a little bit more explicit about the homotopy groups of $BP\\R \\langle n \\rangle$ in the cases $n=1$ and $2$ in Part \\ref{part:LocalCohomology}. \n\n\\subsection{Forms of $BP\\mathbb{R}\\langle n\\rangle$}Our next goal is\nto identify certain spectra as forms of $BP\\R \\langle n \\rangle$. We take the following definition from \\cite{HM}:\n\n\\begin{defn}\nLet $E$ be an even $2$-local commutative and associative $\\Ctwo$-ring spectrum up to homotopy. By \\cite[Lemma 3.3]{HM}, $E$ has a Real orientation and after choosing one, we obtain a formal group law on $\\pi_{*\\rho}^{\\Ctwo}E$. The $2$-typification of this formal group law defines a map $\\pi^e_{2*}BP \\cong \\pi_{*\\rho}^{C_2}BP\\R \\to \\pi_{*\\rho}^{C_2}E$. We call $E$ a \\emph{form of $BP\\R\\langle n\\rangle$} if the map\n\\[\\underline{\\Z_{(2)}[\\vb_1,\\dots, \\vb_n]} \\subset \\underline{\\pi}_{*\\rho}BP\\R \\to \\underline{\\pi}_{*\\rho} E\\]\nis an isomorphism of constant Mackey functors. \n\nThis depends neither on the choice of $\\vb_i$ nor on the chosen Real orientation, as can be seen using that $\\vb_i$ is well-defined modulo $(2, \\vb_1,\\dots, \\vb_{i-1})$. \n\\end{defn}\n\nEquivalently, one can say that $E$ is a form of $BP\\R \\langle n \\rangle$ if and only if\n$E$ is strongly even and its underlying spectrum is a form of\n$BP\\langle n\\rangle$. We want to show that every form of $BP\\R\\langle\nn\\rangle$ is also of the form $BP\\R \/\\vb_{n+1}, \\vb_{n+2}, \\ldots $\nfor some choice of elements $\\vb_i$. For this, we need the following lemma from \\cite[Lemma 3.4]{HM}:\n\\begin{lemma}\\label{lem:regrep}\nLet $f\\colon E\\to F$ be a map of $\\Ctwo$-spectra. Assume that f induces isomorphisms \n\\[\\pi^{\\Ctwo}_{k\\rho}E \\to \\pi^{\\Ctwo}_{k\\rho}E \\quad \\text{and} \\quad \\pi_kE \\to \\pi_kF\\]\nfor all $k\\in\\Z$. Assume furthermore that $\\pi^{\\Ctwo}_{k\\rho-1}E \\to \\pi^{\\Ctwo}_{k\\rho-1}F$ is an injection for all $k\\in\\Z$ (for example, if $\\pi^{\\Ctwo}_{k\\rho-1}E =0$). Then $f$ is an equivalence of $\\Ctwo$-spectra.\n\\end{lemma}\n\\begin{prop}\\label{prop:FormBPRn}\n Let $E$ be a form of $BP\\R\\langle n\\rangle$. Then one can choose\n indecomposables $\\vb_i\\in \\pi_{(2^i-1)\\rho}^{\\Ctwo}BP\\R$ for $i\\geq\n n+1$ such that $E \\simeq BP\\R \/(\\vb_{n+1},\\vb_{n+2},\\dots)$. \n\\end{prop}\n\\begin{proof}\n First choose any system of $\\vb_i$. Choose furthermore a Real orientation $f\\colon BP\\R \\to E$ and denote $f(\\vb_i)$ by $x_i$. Define a multiplicative section \n $$s\\colon \\pi_{*\\rho}^{\\Ctwo}E \\to \\pi_{*\\rho}^{\\Ctwo}BP\\R$$ by $s(x_i) = \\vb_i$ for $1\\leq i \\leq n$. \n \n Now define a new system of $\\vb_i$ by \n \\[\\vb_i^{\\mathrm{new}} = \\vb_i - s(f_*(\\vb_i))\\]\n for $i\\geq n+1$. As these agree with $\\vb_i$ mod $(\\vb_1,\\dots, \\vb_n)$, they are still indecomposable. Furthermore, the $\\vb_i^{\\mathrm{new}}$ are for $i\\geq n+1$ clearly in the kernel of $f_*$. Thus, we obtain a map $BP\\R \\langle n \\rangle\/(\\vb_{n+1}^{\\mathrm{new}},\\vb_{n+2}^{\\mathrm{new}},\\dots) \\to E$ that is an isomorphism on $\\pi_{*\\rho}^{\\Ctwo}$. By Corollary \\ref{Cor:crucial}, the source is strongly even. By Lemma \\ref{lem:regrep}, the map is an equivalence. \n\\end{proof}\n\n\\begin{examples}\\label{exa:forms}We consider Real versions of the classical examples $ku$ and $tmf_1(3)$.\n \\begin{enumerate}\n \\item The connective Real K-theory spectrum $k\\R_{(2)}$ is a form of $BP\\R\\langle 1\\rangle$. Indeed, the underlying spectrum $ku_{(2)}$ is well known to be a form of $BP\\langle 1\\rangle$ and $k\\R_{(2)}$ is also strongly even (as can be seen by the results from \\cite[3.7D]{B-G10} or from the computation in Section \\ref{sec:kRlcss}). \n \\item Define $\\overline{tmf_1(3)}$ as the equivariant connective cover of the spectrum $\\overline{Tmf_1(3)}$, i.e.\\ $Tmf_1(3)$ with the algebro-geometrically defined $\\Ctwo$-action (see \\cite[Section 4.1]{HM} for details). As shown in \\cite[Corollary 4.17]{HM}, $\\overline{tmf_1(3)}_{(2)}$ is a form of $BP\\R\\langle 2\\rangle$. By Proposition \\ref{prop:FormBPRn}, we can construct $\\overline{tmf_1(3)}_{(2)}$ by killing a sequence $\\vb_2, \\vb_3,\\dots$ in $BP\\R$. This construction is used in \\cite{L-OString} to define a $\\Ctwo$-equivariant version of $tmf_1(3)_{(2)}$. In particular, we see (using the discussion before Proposition 4.23 in \\cite{HM}) that $\\overline{TMF_1(3)}_{(2)}$ (with the algebro-geometrically defined $\\Ctwo$-action) agrees with the $\\mathbb{TMF}_1(3)_{(2)}$ of \\cite{L-OString}.\n \\end{enumerate}\n\\end{examples}\n\n\\section{Results and consequences}\\label{sec:results}\nIn this section, we want to discuss our main results in more detail than in the introduction and we will also derive some consequences and give some examples. Recall to that purpose the notation from Sections \\ref{sec:Koszul} and \\ref{sec:BPRBasics}. Furthermore, we will implicitly localize everything at $2$ so that $\\Z$ means $\\Z_{(2)}$ etc. Our main theorem is the following:\n\\begin{thm}\\label{thm:main}\nLet $(m_1,m_2,\\dots)$ be a sequence of nonnegative integers with only finitely many entries bigger than $1$ and let $M$ be the quotient $BP\\R\/(\\vb_1^{m_1},\\vb_2^{m_2},\\dots)$, where we only quotient by the positive powers of $\\vb_i$. Denote by $\\underline{\\vb}$ the sequence of $\\vb_i$ in $\\pi^{\\Ctwo}_{\\bigstar}M\\R$ such that $m_i = 0$, by $|\\underline{\\vb}|$ the sum of their degrees and by $m'$ the sum of all $(m_i-1)|\\vb_i|$ for $m_i> 1$. Then \n$$\\Z^M \\simeq \\Sigma^{-m'+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}; M).$$\n\nThe most important case is that $m_{n+1} = m_{n+2} = \\cdots = 1$ so that \n$$M = BP\\R \\langle n \\rangle\/(\\vb_1^{m_1},\\dots,\\vb_n^{m_n}).$$\nIf $k$ is the number of elements in $\\underline{\\vb}$, we also get \n\\begin{align*}\n\\Z^M \\simeq \\Sigma^{-m'+k+|\\underline{\\vb}| +4-2\\rho}\\Gamma_{\\underline{\\vb}}M,\n\\end{align*}\nwhere we view $M$ as an $M\\R$-module. \n\\end{thm}\nThe first form will be proved as Theorem \\ref{Thm:QuotientDuality} and the second follows from it using Lemma \\ref{lem:Koszul}. The second form also follows from Corollary \\ref{cor:BPRnGorDdish} (using that $\\Gamma_{\\underline{\\vb}}$ preserves cofibre sequences to pass to quotients of $BP\\R \\langle n \\rangle$). \n\n\\begin{example}\\label{ex:BPRn}\n$\\Z^{BP\\R \\langle n \\rangle} \\simeq \\Sigma^{n+D_n\\rho +4-2\\rho}\\Gamma_{(\\vb_1,\\dots, \\vb_n)}BP\\R \\langle n \\rangle$ for $D_n = |v_1|+\\cdots +|v_n|$. This says that $BP\\R \\langle n \\rangle$ has Gorenstein duality with respect to $H\\Zu \\simeq BP\\R \\langle n \\rangle\/(\\vb_1,\\dots, \\vb_n)$. (The last equivalence follows from Corollary \\ref{cor:reduction}.)\n\\end{example}\n\n\\begin{example}\nSet $k\\R(n) = BP\\R \\langle n \\rangle\/(\\vb_1,\\dots, \\vb_{n-1})$ to be connective\nintegral Real Morava $K$-theory and $K\\R(n) = k\\R(n)[\\vb_n^{-1}]$ its periodic version. Then \n\\begin{align*}\\Z^{k\\R(n)} &\\simeq \\Sigma^{1+|\\vb_n|+4-2\\rho}\\Gamma_{\\vb_n}k\\R(n)\\\\\n&\\simeq \\Sigma^{(2^n-3)\\rho+4}\\cof(k\\R(n) \\to K\\R(n))\\end{align*}\nThis includes for $n=1$ the case of usual ($2$-local) connective Real K-theory. \n\\end{example}\n\\begin{example}\n To have a slightly stranger example, take $M = BP\\R\\langle 3\\rangle\/(\\vb_1^4, \\vb_3^2)$. Then \n $$\\Z^M \\simeq \\Sigma^{5-9\\rho}\\Gamma_{\\vb_2}M.$$\n\\end{example}\n\n\\vspace*{0.5cm}\n\nSo far, we have only talked about \\emph{quotients} of $BP\\R$. This does not include important Real spectra like the Real Johnson--Wilson theories $E\\R(n) = BP\\R \\langle n \\rangle[\\vb_n^{-1}]$ or the (integral) Real Morava K-theories $K\\R(n)$. For this, we have to study the behaviour of our constructions under localizations. \n\nLet $M$ be an $RO(C_2)$-graded $\\Z[v]$-module, where $v$ has some degree $|v| \\in RO(C_2)$. We say that $M$ has \\emph{bounded $v$-divisibility} if for every degree $a+b\\sigma$, there is a $k$ such that \n$$v^k\\colon M_{a+b\\sigma-|v^k|} \\to M_{a+b\\sigma}$$\nis zero. We will also apply the concept to modules that are just $\\Z|v|$-graded.\n\\begin{lemma}The class of $RO(C_2)$-graded $\\Z[v]$-modules of bounded $v$-divisibility is closed under submodules, quotients and extensions. \n\\end{lemma}\n\\begin{proof}\n This is clear for submodules and quotients. Let \n $$0 \\to K \\to M \\to N \\to 0$$\n be a short exact sequence of $\\Z[v]$-modules where $K$ and $N$ are of bounded $v$-divisibility. For a given degree $\\alpha \\in RO(C_2)$, we know that there is a $k$ such that $v^k$ maps trivially into $K_\\alpha$. Furthermore, there is an $n$ such that $v^n$ maps trivially into $N_{\\alpha-k|v|}$. Thus, multiplication by $v^{n+k}$ is the zero map $M_{\\alpha-(k+n)|v|} \\to M_\\alpha$.\n\\end{proof}\n\n\nLet $M$ be an $M\\R$-module. We say that $M$ is of \\emph{bounded $\\vb_n$-divisibility} if both $\\pi^{\\Ctwo}_{\\bigstar}M$ and $\\pi^e_*M$ are of bounded $\\vb_n$-divisibility. This is, for example, true if $M$ is connective. \n\n\\begin{lemma}\\label{lem:boundedrho}We have the following two properties of $\\vb_n$-divisiblity.\n\\begin{enumerate}\n\\item Being of bounded $\\vb_n$-divisibility is closed under cofibres and suspensions. \n\\item An $M\\R$-module $M$ is of bounded $\\vb_n$-divisibility if and only if $\\pi^{\\Ctwo}_{*\\rho}M$ and $\\pi^e_*M$ are of bounded $\\vb_n$-divisibility. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nBoth statements follow from the last lemma. For the second item, we additionally use the exact sequence\n$$\\pi^e_{a+b+1}M \\to \\pi^{C_2}_{a+(b+1)\\sigma}M \\to \\pi^{C_2}_{a+b\\sigma}M \\to \\pi^e_{a+b}M$$\ninduced by the cofibre sequence\n$$(C_2)_+ \\to S^0\\to S^{\\sigma}.\\qedhere$$\n\\end{proof}\n\n\\begin{lemma}\nIf $M$ has bounded $\\vb_n$-divisibility, then there is a natural equivalence \n$$M[\\vb_n^{-1}] \\simeq \\Sigma \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits\\left( \\cdots \\to \\Sigma^{|\\vb_n|}\\Gamma_{\\vb_n}M \\xrightarrow{\\vb_n} \\Gamma_{\\vb_n} M\\right)$$\nof $M\\R$-modules.\n\\end{lemma}\n\\begin{proof}\nWe apply the endofunctor $H\\colon N\\mapsto \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits(\\cdots \\to \\Sigma^{|\\vb_n|}N \\xrightarrow{\\vb_n} N)$ of $M\\R$-modules to the cofibre sequence\n$$\\Gamma_{\\vb_n}M \\to M\\to M[\\vb_n^{-1}].$$\nClearly $H(M[\\vb_n^{-1}])\\simeq M[\\vb_n^{-1}]$. Thus, we just have to show that $H(M)\\simeq 0$. This follows by the $\\lim^1$-sequence and bounded $\\vb_n$-divisibility. \n\\end{proof}\n\n\\begin{lemma}\nLet $B$ be a quotient of $BP\\R$ by powers of the $\\vb_i$. Then\n$B[\\vb^{-1}]$ has bounded $\\vb_n$-divisibility if $\\vb$ is a product\nof $\\vb_i$ not containing $\\vb_n$. Hence, the same is also true for\nthe stable Koszul complex $\\Gamma_{\\underline{\\vb}}B$, where $\\underline{\\vb}$ is a sequence of $\\vb_i$ not containing $\\vb_n$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{lem:boundedrho}, it is enough to check the first statement on $\\pi_{*\\rho}^{C_2}$ and on $\\pi^e_*$. On the latter, it is clear and the former is isomorphic to it by Corollary \\ref{Cor:crucial}. For the second statement we use that $\\Gamma_{\\underline{\\vb}}B$ is the fibre of $B \\to \\check{C}(\\underline{\\vb};B)$, where $\\check{C}(\\underline{\\vb};B)$ has a filtration with subquotients $M\\R$-modules of the form $\\Sigma^?B[x^{-1}]$ for some $x\\in \\pi_{\\bigstar}^{C_2}M\\R$ \\cite[Lemma 3.7]{G-M95}. Thus, the second statement follows from Lemma \\ref{lem:boundedrho}. \n\\end{proof}\n\n\\begin{thm}\nLet the notation be as in Theorem \\ref{thm:main} and assume for simplicity that only finitely many $m_i$ are zero and that $m_n = 0$. Then \n$$\\Z^{M[\\vb_n^{-1}]} \\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} M.$$\nHere $\\underline{\\vb} \\setminus \\vb_n$ denotes the sequence of all $\\vb_i$ such that $m_i = 0$ and $i\\neq n$. \n\\end{thm}\n\\begin{proof}\nThe preceding lemmas imply the following chain of equivalences:\n\\begin{align*}\n\\Z^{M[\\vb_n^{-1}]} &\\simeq \\Z^{\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits (M \\xrightarrow{\\vb_n} \\Sigma^{-|\\vb_n|}M \\xrightarrow{\\vb_n} \\cdots)} \\\\\n&\\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits (\\cdots \\xrightarrow{\\vb_n} \\Z^M) \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+k+4-2\\rho}\\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{\\vb_n} \\Gamma_{\\underline{\\vb}}M\\right)\\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+k+4-2\\rho}\\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{\\vb_n} \\Gamma_{\\vb_n}(\\Gamma_{\\underline{\\vb}\\setminus \\vb_n}M) \\right) \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}(\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} M)[\\vb_n^{-1}] \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} (M[\\vb_n^{-1}])\n\\end{align*}\n\\end{proof}\n\n\\begin{example}\nWe recover the following result by Ricka \\cite{Ricka}: \n$$\\Z^{K\\R(n)} \\simeq \\Sigma^{4-2\\rho}K\\R(n).$$\nHere, $K\\R(n)$ denotes integral Morava K-theory $E\\R(n)\/(\\vb_1,\\dots, \\vb_{n-1})$. \n\\end{example}\n\n\\begin{example}\nIn the following, we will use that there are invertible classes $x,\\vb_n\\in\\pi_\\bigstar^{\\Ctwo}E\\R(n)$ of degree $-2^{2n+1}+2^{n+2}-\\rho$ and $(2^n-1)\\rho$ respectively, where $x = \\vb_n^{1-2^n}u^{2^n(1-2^{n-1})}$.\n\\begin{align*}\n\\Z^{E\\R(n)} &\\simeq \\Sigma^{D_{n-1}\\rho + (n-1)+4-2\\rho} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n) \\\\\n&\\simeq \\Sigma^{-(n+2)\\rho+(n+3)} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n) \\\\\n&\\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n).\n\\end{align*}\nThis says that $E\\R(n)$ has Gorenstein duality with respect to $E\\R(n)\/(\\vb_1,\\dots, \\vb_{n-1}) = K\\R(n)$. Note that we can replace the ideal $(\\vb_1,\\dots, \\vb_{n-1})$ by an ideal generated in integral degrees, namely $(\\vb_1x, \\dots, \\vb_{n-1}x^{2^{n-1}-1})$. \n\\end{example}\n\n\\begin{example}\\label{exa:tmf}\nRecall from \\cite{HM} the spectra $tmf_1(3)$, $Tmf_1(3)$ and $TMF_1(3)$ and the corresponding $C_2$-spectra $\\overline{tmf_1(3)}$, $\\overline{Tmf_1(3)}$ and $\\overline{TMF_1(3)}$. Recall that we have $\\pi_*tmf_1(3) = \\Z[a_1,a_3]$, where $a_1$ and $a_3$ can be identified with the images of the Hazewinkel generators $v_1$ and $v_2$, and that $\\overline{tmf_1(3)}$ is a form of $BP\\R\\langle 2\\rangle$ (as already discussed in Example \\ref{exa:forms}). This gives the Anderson dual of $\\overline{tmf_1(3)}$. Tweaking the last theorem a little bit, allows also to show that \n$$\\Z^{\\overline{TMF_1(3)}} \\simeq \\Sigma^{5+2\\rho}\\Gamma_{\\vb_1} \\overline{TMF_1(3)}.$$\nWe can also recover one of the main results of \\cite{HM}, namely that $\\Z^{\\overline{Tmf_1(3)}} \\simeq \\Sigma^{5+2\\rho} \\overline{Tmf_1(3)}$. Indeed, $Tmf_1(3)$ is by \\cite[Section 4.3]{HM} the cofibre of the map \n$$\\Gamma_{\\vb_1,\\vb_2}\\overline{tmf_1(3)} \\to \\overline{tmf_1(3)}.$$\nAs the source is equivalent to $\\Sigma^{-6-2\\rho}\\Z^{\\overline{tmf_1(3)}}$, applying Anderson duality shows that $\\Z^{\\overline{Tmf_1(3)}}$ is the fibre of\n$$\\Sigma^{6+2\\rho}\\overline{tmf_1(3)} \\to \\Sigma^{6+2\\rho} \\Gamma_{\\vb_1,\\vb_2} \\overline{tmf_1(3)}.$$\nThis is equivalent to $\\Sigma^{5+2\\rho}\\overline{Tmf_1(3)}$. \nThis example does not require 2-localization, only that $3$ is inverted.\n\\end{example}\n\n\\begin{remark}By Proposition \\ref{prop:AndersonFixed}, all the results in this section have direct implications for the Anderson duals of the fixed point spectra. These are easiest to understand in the case of $ER(n) = (E\\R(n))^{C_2}$, where we get\n$$\\Z^{ER(n)} \\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3} \\Gamma_{(\\vb_1x,\\dots, \\vb_{n-1}x^{2^n-1})}ER(n).$$\n\\end{remark}\n\n\\vspace{0.8cm}\n\\part{The Gorenstein approach}\nIn this part, we explain the Gorenstein approach to prove Gorenstein duality, first for $k\\R$ and then for $BP\\R \\langle n \\rangle$.\n\\section{Connective $K$-theory with Reality}\n\\label{sec:kR}\nThe present section considers $K$-theory with reality, which is more\nfamiliar than $BP\\R \\langle n \\rangle$ for general $n$, and no 2-localization is\nnecessary. The arguments are especially\nsimple, firstly because $k\\R$ is a commutative ring spectrum, and\nsecondly becaue we only need to\nconsider principal ideals. Simple as the argument is, we see in\nSection \\ref{sec:kRlcss} that the consequences for coefficient rings\nare interesting. \n\n\\subsection{Gorenstein condition and Matlis lift}\nIt is well known that there is a cofibre sequence\n$$\\Sigma^{\\eps}ku\\stackrel{v}\\longrightarrow ku \\longrightarrow H\\Z. $$\nIf one knows the coefficient ring $ku_*=\\Z [v]$, this is easy \nto construct, since we can identify $ku\/v$ as the Eilenberg-MacLane\nspectrum from its homotopy groups. \n\nThere is a version with Reality \\cite{Dugger}. Indeed, we may\nconstruct the cofibre sequence\n$$\\Sigma^{\\rho} k\\R \\stackrel{\\vb}\\longrightarrow k\\R \\longrightarrow H\\Zu, $$\nwhere $k\\R \/\\vb$ is identified using Corollary \\ref{cor:characterizingZ}\n \nSince the Dugger sequence is self dual we immediately deduce that\n$k\\R$ is Gorenstein. \n\n\\begin{lemma}\n\\label{lem:kRGor}\n$$\\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)=\\Sigma^{-\\rho-1}H\\Zu$$\nand $k\\R \\longrightarrow H\\Zu$ is Gorenstein. \n\\end{lemma}\n\n\\begin{proof}\nApply $\\mathrm{Hom}_{k\\R}(\\cdot , k\\R)$ to the Dugger sequence. \n\\end{proof}\n\nTo actually get Gorenstein duality we need to construct a Matlis\nlift (adapted from \\cite[Section 6]{DGI}), which is a counterpart in topology of the injective hull of the residue\nfield. \n\\begin{defn}\nIf $M$ is an $H\\Zu$-module, we say that a $k\\R$-module $\\tilde{M}$ is\na {\\em Matlis lift} of $M$ if $\\tilde{M}$ is $H\\Zu$-$\\R$-cellular and \n$$\\mathrm{Hom}_{k\\R}(T, \\tilde{M})\\simeq \\mathrm{Hom}_{H\\Zu}(T, M)$$\nfor all $H\\Zu$-modules $T$.\n\\end{defn}\n\nThe Anderson dual provides one such example. \n\n\\begin{lemma}\n\\label{lem:ML}\nThe $k\\R$-module $\\Sigma^{-2(1-\\sigma)}\\Z^{k\\R}$ is a Matlis lift of\n$H\\Zu$. Indeed, \n\n(i) $\\underline{\\Z}^{k\\R}$ is $H\\Zu$-$\\R$-cellular and \n\n(ii) There is an equivalence \n$$\\Sigma^{2\\pp} H\\Zu\\simeq H\\Zu^*=\\mathrm{Hom}_{k\\R} (H\\Zu, \\Z^{k\\R}), $$\nwhere $\\delta =1-\\sigma$. \n\\end{lemma}\n\n\\begin{proof}\nOne could prove the first part from the slice tower, but it also follows directly from Corollary \\ref{cor:cellular}. \n\nThe second statement is immediate from Lemma \\ref{lem:Zu}. \n\\end{proof}\n\n\\subsection{Gorenstein duality}\n We next want to move on to Gorenstein duality, so we write\n$$\\cE=\\mathrm{Hom}_{k\\R}(H\\Zu, H\\Zu). $$\n\nCombining Lemmas \\ref{lem:kRGor} and \\ref{lem:ML}, we have \n\\begin{eqnarray}\\label{eq:kReq}\\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)\\simeq \\Sigma^{-\\rho-1}H\\Zu \\simeq\n\\mathrm{Hom}_{k\\R}(H\\Zu, \\Sigma^{-4+\\sigma}\\Z^{k\\R})\\end{eqnarray}\n\nWe now want to remove the $\\mathrm{Hom}_{k\\R}(H\\Zu , \\cdot )$ from this\nequivalence. \n\n\\begin{lemma} {\\em (Effective constructibility)}\n\\label{lem:effective}\nThe evaluation map \n$$\\mathrm{Hom}_{k\\R}(H\\Zu, M)\\otimes_{\\cE}H\\Zu \\longrightarrow M$$\nis $H\\Zu$-$\\R$-cellularization for every left $k\\R$-module $M$.\n\\end{lemma}\n\n\\begin{proof}\nSince the domain is clearly $H\\Zu$-$\\R$-cellular, it is enough to show the map is an\nequivalence for all cellular modules $M$.\n\nThis is clear for $M=H\\Zu$. The class of $M$ for which the statement is true is closed under (i) triangles, (ii)\ncoproducts (since $H\\Zu$ is small) and (iii) suspensions by\nrepresentations. This gives all $\\R$-cellular modules. \n\\end{proof}\n\nLocal cohomology gives an alternative approach to\ncellularization. Recall that we define the $\\vb$-power torsion of a $k\\R$-module $M$\nby the fibre sequence\n$$\\Gamma_{\\vb}M \\longrightarrow M \\longrightarrow M[1\/\\vb]. $$\n\nThe following lemma is a special case of Proposition \\ref{prop:cell}. \n\\begin{lemma}\n\\label{lem:Gammacell}\nThe map \n$$\\Gamma_{\\vb}M\\longrightarrow M$$ \nis $H\\Zu$-$\\R$-cellularization.\n\\end{lemma}\n\nIt remains to check that the two $\\cE$-actions on $H\\Zu$ coincide. \n\n\n\\begin{lemma}\\label{lem:UniquenesskR}\nThere is a unique right $\\cE$-module structure on $H\\Zu$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $H\\Zu'$ is a right $\\cE$-module whose underlying\n$C_2$-spectrum is equivalent to the\nEilenberg-MacLane spectrum $H\\Zu$. \nWe first claim that $H\\Zu'$ can be constructed as an\n$\\cE$-module with cells in degrees $k\\rho$ for $k\\leq 0$:\n$$H\\Zu'\\simeq_{\\cE} S^0_{\\cE}\\cup e^{-\\rho}_{\\cE}\\cup e^{-2\\rho}_{\\cE} \\cup \\cdots $$\n\nOnce that is proved, we argue as follows. If $H\\Zu''$ is another right\n$\\cE$-module with underlying $C_2$-spectrum $H\\Zu$, we may construct a\nmap $H\\Zu'\\longrightarrow H\\Zu''$ skeleton by skeleton in the usual way. \nWe start with the $\\cE$-module map $\\cE =(H\\Zu')^{(0)}\\longrightarrow H\\Zu'$ giving\nthe unit, and successively extend the map over the\ncells of $H\\Zu'$. At each stage the obstruction to the existence of an \nextension over $(H\\Zu')^{-k\\rho}$ lies in $\\pi^{\\Ctwo}_{-k\\rho-1}(H\\Zu'')$. \nThese groups are zero. We end with a map which is an isomorphism on \n0th homotopy Mackey functors and therefore an equivalence.\n\n\nFor the cell-structure, it is enough to show that for every right\n$\\cE$-module $H\\Zu'$ of the homotopy type of the Eilenberg--MacLane\nspectrum $H\\Zu$, there is a map $\\cE \\to H\\Zu'$ of right $\\cE$-modules\nwhose fibre has the homotopy type of $\\Sigma^{-\\rho-1}H\\Zu$. Indeed, suppose we have already constructed a right $\\cE$-module $(H\\Zu')^{(n)}$ with an $\\cE$-map to $H\\Zu'$ with fibre of the homotopy type $\\Sigma^{-(n+1)\\rho-1}H\\Zu$. Then it is easy to see that the cofibre $(H\\Zu')^{(n+1)}$ of the map $\\Sigma^{-(n+1)\\rho -1}\\cE \\to \\Sigma^{-(n+1)\\rho-1}H\\Zu \\to (H\\Zu')^{(n)}$ has the analogous property. Taking the homotopy colimit, we get a map $\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits (H\\Zu')^{(n)} \\to H\\Zu'$ with fibre $\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{-(n+1)\\rho-1}H\\Zu$, which is clearly zero (e.g.\\ by Lemma \\ref{lem:regrep} and the fact that $H\\Zu$ is even; we refer to \\cite[Section 3.4]{Ricka} for a table of $\\underline{\\pi}_{\\bigstar}^{C_2}H\\Zu$). \n\nWe choose the map $f\\colon \\cE \\to H\\Zu'$ representing $1\\in \\pi_0^{\\Ctwo}H\\Zu'$ and call the fibre $F$. We want to show that $f$ agrees with the canonical map $\\cE \\to H\\Zu$ on homotopy groups of the form $\\pi_{k-\\sigma}^{\\Ctwo}$ for $k\\in\\Z$. Indeed, the only nonzero class in $H\\Zu'$ in these degrees is $a\\in\\pi_{-\\sigma}^{\\Ctwo}H\\Zu'$, which has to be hit by $a\\in \\pi_{-\\sigma}^{\\Ctwo}\\cE$ as it comes from the sphere. Thus, $\\pi_{k-\\sigma}^{\\Ctwo} F \\cong \\pi_{k-\\sigma}^{\\Ctwo} \\Sigma^{-1-\\rho}H\\Zu$ for all $k$ and hence $F\\simeq \\Sigma^{-1-\\rho}H\\Zu$ as $C_2$-spectra, as we needed to show. \n\\end{proof}\n\nFrom this the required statement follows. \n\n\\begin{cor} {\\em (Gorenstein duality)} \n\\label{cor:kRGorD}\nThere is an equivalence of $k\\R$-modules\n$$\\Gamma_{\\vb} k\\R \\simeq \\Sigma^{-4+\\sigma} \\Z^{k\\R}. \\qed \\\\[1ex]$$\n\\end{cor}\n\\begin{proof}\n By \\eqref{eq:kReq} and Lemma \\ref{lem:UniquenesskR}, we know that \n $$ \\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)\\otimes_{\\cE} H\\Zu\\simeq \n\\mathrm{Hom}_{k\\R}(H\\Zu, \\Sigma^{-4+\\sigma}\\Z^{k\\R})\\otimes_{\\cE}H\\Zu.$$\nBy Lemma \\ref{lem:effective}, the two sides are the cellularizations of $k\\R$ and $\\Sigma^{-4+\\sigma}\\Z^{k\\R}$ respectively. By Lemmas \\ref{lem:Gammacell} and \\ref{lem:ML}, the former is $\\Gamma_{\\vb}k\\R$ and the latter is $\\Sigma^{-4+\\sigma}\\Z^{k\\R}$ itself. \n\\end{proof}\n\n\nThe implications of this equivalence for the coefficient ring are\ninvestigated in Section \\ref{sec:kRlcss}. \n\n\\section{\\texorpdfstring{$\\protect BP\\langle n \\rangle$}{BP} with Reality}\n\\label{sec:dishonest}\n\nWe now turn to the case of $BP\\R \\langle n \\rangle$ for a general $n$. The counterpart\nof the argument of Section \\ref{sec:kR} is a little simpler when $BP\\R \\langle n \\rangle$ is a commutative ring\nspectrum. For $n=1$ and $n=2$, the spectra $k\\R$, and $tmf_1(3)$, are both known to be a\ncommutative ring spectra, and their 2-localizations give $BP\\R \\langle n \\rangle$\nwhen $n=1$ and $n=2$ respectively. However for higher\n$n$ it is not known that $BP\\R \\langle n \\rangle$ is a commutative ring spectrum. \nThis is a significant technical issue, but\none that is familiar when working with non-equivariant $BP$-related theories\nsince $BP$ is not known to be a commutative ring. The established\nmethod for getting around this is to use the fact that \n$BP$ and $BP\\langle n \\rangle$ are modules over the commutative ring $MU$.\nWe will adopt precisely the same method by working with $M\\R$-modules.\nThe only real complication is that \nwe are forced to work with spectra whose homotopy groups are bigger\nthan we might like, but if we focus on the relevant part, it causes\nno real difficulties. \n\n\n\n\n\n\\subsection{Gorenstein condition and Matlis lift}\nAs mentioned in the introduction of this section, we will work in the setting of $M\\R$-modules. More precisely, we will always (implicitly) localize at $2$ and set $S = M\\R_{(2)}$. As discussed in Section \\ref{sec:BPRBasics}, we can define $S$-modules $BP\\R \\langle n \\rangle$, once we have chosen a sequence of $\\vb_i$ (for example, the Hazewinkel or Araki generators). \n\nThe ideal \n$$\\Jb_n=(\\vbn{1}, \\ldots, \\vbn{n})$$\nplays a prominent role, and we will abuse notation by writing \n$$S\/\\overline{J}_n:=\\cof(S\\stackrel{\\vbn{1}}\\longrightarrow S)\\otimes_S\n\\cof(S\\stackrel{\\vbn{2}}\\longrightarrow S)\\otimes_S \\cdots \\otimes_S\n\\cof(S\\stackrel{\\vbn{n}}\\longrightarrow S), $$\nand then \n$$M\/\\overline{J}_n:=M\\otimes_S S\/\\overline{J}_n.$$\nIn particular, \n$$BP\\R \\langle n \\rangle\/\\overline{J}_n=BP\\R \\langle n \\rangle\/\\vbn{n}\/\\vbn{n-1}\/\\cdots \/\\vbn{1}\\simeq H\\Zu $$\nby the $C_2$-case of the reduction theorem, here proved as Corollary \\ref{cor:reduction}. \n\nIf $BP\\R \\langle n \\rangle$ is a ring spectrum\n$$\\mathrm{Hom}_{BP\\R \\langle n \\rangle}(H\\Zu, M)=\\mathrm{Hom}_{BP\\R \\langle n \\rangle}(BP\\R \\langle n \\rangle\\otimes_S S\/\\overline{J}_n,\nM)=\\mathrm{Hom}_{S}(S\/\\overline{J}_n, M), $$\nso that the right hand side gives a way for us to express the\nfact that certain $BP\\R \\langle n \\rangle$-modules (such as $BP\\R \\langle n \\rangle$ and $\\Z^{BP\\R \\langle n \\rangle}$)\nare Matlis lifts, using only module structures over $S$.\n\nApplying this when $M=BP\\R \\langle n \\rangle$, we obtain the Gorenstein condition. \n\n\\begin{lemma}\n\\label{lem:BPRnGordish}\nThe map $BP\\R \\langle n \\rangle \\longrightarrow H\\Zu$ is Gorenstein of shift $-D_n\\rho -n$ \nin the sense that \n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, BP\\R \\langle n \\rangle)\\simeq \\Sigma^{-D_n\\rho -n}H\\Zu, $$\nwhere\n$$D_n\\rho =|\\vbn{n}|+|\\vbn{n-1}|+\\cdots\n+|\\vbn{1}|=\\left[2^{n+1}-n-2 \\right]\\rho . $$\n\\end{lemma}\n\n\\begin{proof}\nSince each of the maps $\\vbn{i}: \\Sigma^{|\\vbn{i}|}S\\longrightarrow S$ is self-dual, \nfor any $S$-module $M$, we have \n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, M)\\simeq \\Sigma^{-D_m\\rho-n}S\/\\overline{J}_n \\otimes_S M. $$\n\\end{proof}\n\nApplying this when $M=\\Z^{BP\\R \\langle n \\rangle}$, we obtain the Anderson Matlis lift.\n\n\\begin{lemma}\n\\label{lem:MLB}\nThe Anderson dual of $BP\\R \\langle n \\rangle$ is a Matlis lift of $H\\Zu^*$ in the sense that \n\n(i) $\\Z^{BP\\R \\langle n \\rangle}$ is $H\\Zu$-$\\R$-cellular and \n\n(ii) There is an equivalence \n$$\\Sigma^{2-2\\sigma}H\\Zu \\simeq H\\Zu^* \\simeq \\mathrm{Hom}_{S} (S\/\\overline{J}_n, \\Z^{BP\\R \\langle n \\rangle}). $$\n\\end{lemma}\n\n\\begin{proof}\nOne could prove the first part from the slice tower, but it also follows directly from Corollary \\ref{cor:cellular}. \n\nFor the second statement observe that \n$$\\mathrm{Hom}_{S} (S\/\\overline{J}_n, \\Z^{BP\\R \\langle n \\rangle}) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n\\otimes_S BP\\R \\langle n \\rangle, \\Z^S)\\simeq \\Z^{H\\Zu}.$$ \nThus, Lemma \\ref{lem:Zu} implies the statement. \n\\end{proof}\n\n\n\n\\subsection{Gorenstein duality}\nThroughout this section, we will write $R = BP\\R \\langle n \\rangle$ for brevity. \nCombining Lemmas \\ref{lem:BPRnGordish} and \\ref{lem:MLB}, we have an\nequivalence of $S$-modules\n$$\\mathrm{Hom}_{S}(S\/\\overline{J}_n, R)\\simeq \\Sigma^{-D_n \\rho-n}H\\Zu \\simeq\n\\mathrm{Hom}_{S}(S\/\\overline{J}_n, \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma}\\Z^R)$$\n\nWe now want to remove the $\\mathrm{Hom}_{S }(S\/\\overline{J}_n , \\cdot )$ from this\nequivalence. The endomorphism ring \n$$\\widetilde{\\cE}_n =\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n)$$\nof the small $S$-module $S\/\\overline{J}_n$, replaces $\\cE_n=\\mathrm{Hom}_{R}(H\\Zu,\n H\\Zu)$ from the case that $R =BP\\R \\langle n \\rangle$ is a ring spectrum. We note that \n$$\\widetilde{\\cE}_n\\otimes_S R=\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n )\\otimes_S R\\simeq \n\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n)\\otimes_S R). $$\nIf $R = BP\\R \\langle n \\rangle$ were a commutative ring, this would be a ring equivalent to\n$\\mathrm{Hom}_R(H\\Zu, H\\Zu)$. \n\n\nIn any case, the following is proved exactly like Lemma \\ref{lem:effective}\n\n\\begin{lemma} {\\em (Effective constructibility)}\\label{lem:effective2}\nThe evaluation map \n$$\\mathrm{Hom}_{S }(S\/\\overline{J}_n, M)\\otimes_{\\widetilde{\\cE}_n}S\/\\overline{J}_n \\longrightarrow M$$\nis $S\/\\overline{J}_n$-$\\R$-cellularization.\\qed \\\\[1ex] \n\\end{lemma}\n\n\n\nOf course local cohomology gives an alternative approach to\ncellularization. Recall that we define\n$$\\Gamma_{\\Jb_n}M =\\Gamma_{\\vbn{1} }S\n\\otimes_{S}\\Gamma_{\\vbn{2}}S \\otimes_{S}\\cdots \\otimes_{S}\n\\Gamma_{\\vbn{n} }S \\otimes_{S}M. $$\nThen Proposition \\ref{prop:cell} gives the following lemma. \n\n\\begin{lemma}\n$$\\Gamma_{\\Jb_n}M\\longrightarrow M$$ \nis $H\\Zu$-$\\R$-cellularization.\n\\end{lemma}\n\nIt remains to check that the two $\\widetilde{\\cE}_n$ actions on $H\\Zu$ coincide. For\n$k\\R$ (i.e., $n=1$) we showed there was a unique right $\\cE_n$-module\nstructure on $H\\Zu$. This may be true for $\\widetilde{\\cE}_n$-module structures, but we will instead\njust prove in the next subsection that the two particular $\\widetilde{\\cE}_n$-modules that\narose from the left and right hand ends of the first display of this subsection are equivalent. \n\nThe required Gorenstein duality statement follows. Its implications for the coefficient ring for\n$n=2$ are investigated explicitly in Section \\ref{sec:tmfotlcss}. \n\n\\begin{cor} \n\\label{cor:BPRnGorDdish}\n{\\em (Gorenstein duality)} There is an equivalence of $M\\R$-modules\n$$\\Gamma_{\\Jb_n} R \\simeq \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma} \\Z^R$$\nwith $R = BP\\R \\langle n \\rangle$.\n\\end{cor}\n\\begin{proof} \nWe will argue in Subsection \\ref{subsec:Eequiv} that the equivalence\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, R) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{-D_n\\rho\n-n-2\\delta} \\Z^R), $$\nis in fact an equivalence of right\nmodules over $\\widetilde{\\mathcal{E}}_n$. By Lemma \\ref{lem:effective2}, $R$ and \n$\\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma}\n\\Z^R$ have equivalent $S\/\\overline{J}_n$ cellularizations. We have seen above that the cellularization of $R$ is $\\Gamma_{\\Jb_n} BP\\R \\langle n \\rangle $ and that $\\Sigma^{-D_n\\rho -n-2\\delta}\n\\Z^R$ itself is cellular. \n\\end{proof}\n\n\n\n\n\\subsection{The equivalence of induced and coinduced Matlis lifts of\n $\\protect H\\Zu$}\n\\label{subsec:Eequiv}\nFor brevity we will still write $R=BP\\R \\langle n \\rangle$, and note that we have a map\n$S=M\\R \\longrightarrow BP\\R \\langle n \\rangle=R$. The two $S$-modules that concern us are of a\nvery special sort, one looks as if it is obtained from an $S$-module\nby `extension of scalars from $S$ to $R$' and one\nlooks as if it is obtained by `coextension of scalars from $S$ to $R$'.\n\n\\begin{lemma}\n\\label{lem:resEtnEn}\nWe have equivalences of right $\\widetilde{\\mathcal{E}}_n$-modules\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, R)\\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, S)\\otimes_SR.$$\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Z^R)=\\mathrm{Hom}_S(R, \\mathrm{Hom}_S(S\/\\overline{J}_n, \\Z^S))$$\n\\end{lemma}\n\n\\begin{proof}\nThe first equivalence is immediate from the smallness of\n$S\/\\overline{J}_n$. \n\nThe second equivalence follows from the equivalence \n$$\\Z^R\\simeq \\mathrm{Hom}_S(R, \\Z^S)$$\nof $S$-modules. \n\\end{proof}\n\n\n\nSuspending the equivalences from Lemma \\ref{lem:resEtnEn} so that we are comparing two $\\widetilde{\\mathcal{E}}_n$-modules\nequivalent to $H\\Zu$ (see Lemma \\ref{lem:MLB}) we have\n$$Y_1=\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{D_n\\rho+n}R)\\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}S)\\otimes_SR=X_1\\otimes_SR$$\nand \n$$Y_2=\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{2\\pp} \\Z^R)\\simeq \\mathrm{Hom}_S(S,\n\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{2\\pp} \\Z^S)) =\\mathrm{Hom}_S(R,X_2). $$\n\n \nIn Subsection \\ref{subsec:alpha} we will construct an $\\widetilde{\\mathcal{E}}_n$-map $\\alpha: X_1\\longrightarrow Y_2$ and then argue\nin Subsection \\ref{subsec:alphat} that this extends along $X_1=X_1\\otimes_SS\\longrightarrow X_1\\otimes_SR =Y_1$ to\ngive a map $\\tilde{\\alpha}: Y_1\\longrightarrow Y_2$ which is easily seen to be an\nequivalence: it is clearly a $*\\rho -$ isomorphism\nand hence an equivalence by Lemma \\ref{lem:regrep}. \n\nTo see our strategy, note that the extension problem \n$$\\diagram\nX_1\\dto \\rto^-{\\alpha} & \\mathrm{Hom}_S(S\/\\overline{J}_n, \\mathrm{Hom}_S(R,\\Z^S))\\\\\nX_1\\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}}&\n\\enddiagram$$\nin the category of $\\widetilde{\\mathcal{E}}_n$-modules is equivalent to the extension problem\n$$\\diagram\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R\n\\dto \\rto^-{\\alpha'} & \\Z^S\\\\\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R\\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}'}&\n\\enddiagram$$\nin the category of $S$-modules. The point is that by the defining property of the Anderson dual, this\nlatter extension problem can be tackled by looking in\n$\\pi^{\\Ctwo}_0$. The 0th homotopy groups of the spectra on the left are\neasily calculated from the known ring $\\pi^{\\Ctwo}_{\\bigstar}(H\\Zu)$. \n\n\\subsection{Construction of the map $\\alpha$}\n\\label{subsec:alpha}\n\nWe construct the map $\\alpha$ using a similar method as in the proof of Lemma \\ref{lem:UniquenesskR}.\n\n\\begin{lemma}\nThere is a map \n$$\\alpha : X_1 \\longrightarrow Y_2$$\nof right $\\widetilde{\\mathcal{E}}_n$-modules that takes the image of $1\\in\n\\pi^{\\Ctwo}_0(S)$ to a generator of $\\pi^{\\Ctwo}_0(H\\Zu)=\\Z$. \n\\end{lemma}\n\n\\begin{proof}\nFirst we claim that $X_1$ has a $\\widetilde{\\mathcal{E}}_n$-cell structures \nwith one 0-cell and other cells in dimensions which are negative\nmultiples of $\\rho$. More precisely, there is \n a filtration \n$$\\widetilde{\\mathcal{E}}_n\\simeq X_1^{[0]}\\to X_1^{[1]}\\to\nX_1^{[2]}\\to \\cdots \\to X_1$$\nso that $X_1\\simeq \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits_dX_1^{[d]}$ and there are cofibre sequences\n$$X_1^{[d-1]}\\longrightarrow X_1^{[d]}\\longrightarrow \\bigvee \\Sigma^{-d\\rho} \\widetilde{\\mathcal{E}}_n. $$\n\nBy definition $X_1 = \\mathrm{Hom}_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}S)$. By Proposition \\ref{prop:cell} and Lemma \\ref{lem:Koszul}, this is equivalent to \n$$Hom_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}\\Gamma_{\\overline{J}_n}S) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, \\kappa_S(\\vb_1,\\dots, \\vb_n))$$\nbecause $\\Gamma_{\\overline{J}_n}S \\to S$ is $S\/\\overline{J}_n$-$\\R$-cellularization. The usual construction of the stable Koszul complex from the unstable\nKoszul complex recalled in Subsection \\ref{sec:Koszul}, shows that \n$$\\kappa_S(\\vb_1,\\dots, \\vb_n)$$ \nhas a filtration with subquotients sums of $(-k\\rho)$-fold suspensions of $S\/\\overline{J}_n$. This induces a corresponding filtration on $X_1$. \n\n\nAs in Lemma \\ref{lem:UniquenesskR} we may construct $\\alpha$ by obstruction\ntheory. Indeed, we start\nby choosing a map $\\widetilde{\\mathcal{E}}_n=X_1^{[0]}\\longrightarrow Y_2^{[0]}$ taking the unit to\na generator. At the $d$th stage we have a\nproblem \n$$\\diagram\nX_1^{[d-1]} \\rto \\dto & Y_2\\\\\nX_1^{[d]} \\ar@{-->}[ur]&\n\\enddiagram$$\nThe obstruction to extension is in a finite product of groups\n$$[\\Sigma^{-d\\rho-1}\\widetilde{\\mathcal{E}}_n, Y_2]^{\\widetilde{\\mathcal{E}}_n}=\\pi^{\\Ctwo}_{-d\\rho -1}(H\\Zu)=0 $$\nwhere the vanishing is from the known value of $\\pi^{\\Ctwo}_{\\bigstar}(H\\Zu)$. \n\\end{proof}\n\n\\subsection{The map $\\tilde{\\alpha}$}\n\\label{subsec:alphat}\nReferring to the second extension problem diagram above, we note $S\/\\overline{J}_n\\otimes_SR\\simeq\nH\\Zu$ as $S$-modules. Thus, we have to solve the lifting problem\n$$\\diagram\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} H\\Zu \\otimes_S S \\dto_{1\\tensor1\\otimes \\pi} \\rto^-{\\alpha'} & \\Z^S\\\\\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} H\\Zu \\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}'}&\n\\enddiagram$$\nwhere $H\\Zu$ is equipped with some $\\widetilde{\\mathcal{E}}_n$-module structure. Denote the upper left corner by $T$. The map $T \\to T\\otimes_S R$ is a split inclusion on underlying $MU$-modules. Indeed, \n$$T\\simeq X_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R$$\n and the map $R \\to R\\otimes_S R$ is a split inclusion on underlying spectra because $BP\\langle n\\rangle$ has the structure of a homotopy unital $MU$-algebra \\cite[V.2.6]{EKMM}. \n\nBy the definition of Anderson duals, we have a diagram of short exact sequences:\n\\[\\xymatrix{\n 0 \\ar[r]& \\mathrm{Ext}_\\Z^1(\\pi^{\\Ctwo}_{-1}(T\\otimes_SR),\\Z)\\ar[d] \\ar[r]& [T \\otimes_SR, \\Z^S]^S \\ar[d]\\ar[r]& \\mathrm{Hom}_\\Z(\\pi^{\\Ctwo}_0(T \\otimes_SR),\\Z)\\ar[d] \\ar[r]& 0 \\\\\n 0\\ar[r]&\\mathrm{Ext}_\\Z^1(\\pi^{\\Ctwo}_{-1}(T),\\Z)\\ar[r]& [T, \\Z^S]^S \\ar[r]& \\mathrm{Hom}_\\Z(\\pi^{\\Ctwo}_0(T),\\Z) \\ar[r]& 0\n }\n\\]\n\nWe want to show that the maps $\\pi_k^{C_2}T \\to \\pi_k^{C_2}T\\otimes_S R$ are split injections for $k=0,-1$, which solves the problem. For the computation of $\\pi_*^{C_2}T$ recall from the last section that $X_1$ has a filtration starting with $X_1^{[x]} = \\widetilde{\\mathcal{E}}_n$ and with subquotients sums of terms of the form $\\Sigma^{-d\\rho}\\widetilde{\\mathcal{E}}_n$. Thus, $T$ obtains a filtration starting with $T^{[1]} = H\\Zu$ and with subquotients sums of terms of the form $\\Sigma^{-d\\rho}H\\Zu$. The map $H\\Zu = T^{[1]} \\to T$ clearly induces isomorphisms on $\\underline{\\pi}_k^{\\Ctwo}$ for $k = 0,-1$ by the known homotopy groups of $H\\Zu$ (see e.g.\\ \\cite[Section 3.4]{Ricka} for a table). Thus, $\\underline{\\pi}^{C_2}_{-1}T = 0$ and $\\underline{\\pi}^{C_2}_0T = \\underline{\\Z}$. \n\nIf we have a map $\\underline{\\Z} \\to M$ from the constant Mackey functor, it is a split injection on $(C_2\/C_2)$ if it is one on $(C_2\/e)$. But we have already seen above that on underlying spectra $T\\to T\\otimes_SR$ is a split inclusion. Thus, we have shown that $\\pi_k^{C_2}T \\to \\pi_k^{C_2}(T\\otimes_S R)$ is split injective, which provides the map $\\tilde{\\alpha}'$.\n\n\\vspace{1cm}\n\\part{The hands-on approach}\nIn this part, we give a different way to compute the Anderson dual of $BP\\R \\langle n \\rangle$ by first computing the Anderson dual of $BP\\R$ itself. Again, we will first do the case of $k\\R$. \n\n\\section{The case of $k\\mathbb{R}$ again}\\label{sec:kRagain}\nTo illustrate our strategy, we give an alternative calculation of the\nAnderson dual of $k\\R$. This can also be deduced from our main theorem\nbelow, but it might be helpful to see the proof in this simpler case\nfirst. General references for the $RO(C_2)$-graded homotopy groups of $k\\R$ are \\cite[Section 3.7]{B-G10} or Section \\ref{sec:kRgroups}.\n\nWe want to show the following proposition:\n\n\\begin{prop}\nThere is an equivalence $\\kappa_{k\\R}(\\vb) \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}.$\n\\end{prop}\nRecall here that $\\vb \\in \\pi^{C_2}_{\\rho}k\\R$ is the Bott element for Real K-theory and \n$$\\kappa_{k\\R}(\\vb) = \\hcolim_n \\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n.$$\n Our idea is simple: To obtain a map from the homotopy colimit, we have just to give maps \n $$\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$$\n that are compatible in the homotopy category (see Remark \\ref{rmk:hocolim}). We will show in the next lemma that these maps are essentially unique: The Mackey functor of homotopy classes of $k\\R$-linear maps $\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$ is isomorphic to $\\underline{\\Z}$ and the precomposition with the map $\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{-n\\rho}k\\R\/\\vb^{n+1}$ induces the identity on $\\underline{\\Z}$. \n\nChoosing the $C_2$-equivariant map $\\kappa_{k\\R}(\\vb) \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$ that corresponds to $1\\in \\Z$ for every $n$ induces an equivalence on underlying homotopy groups. By Lemma \\ref{lem:regrep} the result follows as soon as we have established that $\\kappa_{k\\R}(\\vb)$ is strongly even and that the Mackey functor $\\underline{\\pi}_{*\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}$ is constant. These two facts will also be shown in the following lemma, finishing the proof of the proposition.\n\n\\begin{lemma}Denote for a $\\underline{\\Z}[\\vb]$ module $M$ by $\\{M\\}_{\\vb^n}$ the $\\vb^n$-torsion in it. Then we have:\n \\begin{enumerate}\n \\item $k\\R\/\\vb^n$ is strongly even and hence the same is true for $\\kappa_{k\\R}(\\vb)$.\n \\item $\\underline{\\pi}_{n\\rho}^{\\Ctwo}\\Sigma^{2\\rho-4}\\Z^{k\\R} \\cong \\underline{\\pi}^{\\Ctwo}_{(n-2)\\rho+4}\\Z^{k\\R}$ is constant for all $n\\in\\Z$.\n \\item $\\underline{[\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n, \\Sigma^{2\\rho-4}\\Z^{k\\R}]}^{C_2}_{k\\R} \\cong \\left\\{\\underline{\\pi}^{\\Ctwo}_{-(n-1)\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}\\right\\}_{\\vb^n} \\cong \\underline{\\Z}$\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n The first part follows as \n $$\\underline{\\pi}^{\\Ctwo}_{k\\rho -i}(k\\R\/\\vb^n) =\\underline{\\pi}^{\\Ctwo}_{k\\rho -i}(k\\R)\/\\vb^n$$\n for $i=0,1$ because $\\pi_{k\\rho-i}^{C_2}k\\R = 0$ for $i=1,2$. \n \n For the second part consider the short exact sequence\n \\[0\\to \\mathrm{Ext}(\\underline{\\pi}^{C_2}_{k\\rho-5}k\\R, \\Z) \\to \\underline{\\pi}^{C_2}_{-k\\rho +4}\\Z^{k\\R} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{k\\rho -4}k\\R,\\Z) \\to 0.\\]\n We have $\\underline{\\pi}^{\\Ctwo}_{k\\rho-5}k\\R = 0$ for all $k\\in\\Z$. For $k<2$, the Mackey functor $\\underline{\\pi}^{\\Ctwo}_{k\\rho-4}k\\R$ vanishes as well and for $k\\geq 2$, we have $\\underline{\\pi}^{\\Ctwo}_{k\\rho-4}k\\R \\cong \\underline{\\Z}^*$, generated by $v^{k-2}$ and $2\\vb^{k-2}u$. Thus, \n $$\\underline{\\pi}^{\\Ctwo}_{-k\\rho+4}\\Z^{k\\R} \\cong \\begin{cases} 0 & \\text{ if }k<2 \\\\\n \\underline{\\Z} & \\text{ if }k\\leq 2 \\end{cases}$$\n This shows part (2). As multiplication by $\\vb^n$ does not hit $\\underline{\\pi}^{\\Ctwo}_{(n+1)\\rho-4}k\\R$, the whole Mackey functor $\\underline{\\pi}^{\\Ctwo}_{-(n+1)\\rho+4}\\Z^{k\\R}$ is $\\vb^n$-torsion. This gives the second isomorphism of the third part. \n \n \n For the remaining isomorphism, note that the cofibre sequence\n \\[\n \\Sigma^\\rho k\\R \\xrightarrow{\\vb^n} \\Sigma^{-(n-1)\\rho}k\\R \\to \\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{\\rho+1} k\\R\n \\]\ninduces a short exact sequence\n \\[0 \\to (\\underline{\\pi}^{\\Ctwo}_{\\rho +1}\\Sigma^{2\\rho-4}\\Z^{k\\R} )\/ \\vb_n \\to \\underline{[\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n, \\Sigma^{2\\rho-4}\\Z^{k\\R}]}^{C_2}_{k\\R} \\to \\left\\{\\underline{\\pi}^{\\Ctwo}_{-(n-1)\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}\\right\\}_{\\vb^n} \\to 0\\]\n \n We have $\\underline{\\pi}_{\\rho +1}^{\\Ctwo}\\Sigma^{2\\rho-4}\\Z^{k\\R} \\cong \\underline{\\pi}_{5 -\\rho}^{\\Ctwo}\\Z^{k\\R}$, which sits in a short exact sequence\n \\[ 0 \\to \\mathrm{Ext}_\\Z(\\underline{\\pi}_{\\rho-6}^{\\Ctwo}k\\R,\\Z) \\to \\underline{\\pi}_{5 -\\rho}^{\\Ctwo}\\Z^{k\\R} \\to \\mathrm{Hom}_\\Z(\\underline{\\pi}_{\\rho-5}^{\\Ctwo}k\\R,\\Z)\\to 0.\\]\n But because of connectivity, $\\underline{\\pi}_{\\rho-c}^{\\Ctwo}k\\R = 0$ for $c\\geq 3$. \n \\end{proof}\n\n\\section{Duality for $BP\\mathbb{R}$}\nWe will use throughout the abbreviation $B = BP\\R$ and will furthermore implicitly localize everything at $2$ so that $\\Z = \\Z_{(2)}$ etc.\\ and all $\\mathrm{Hom}$ and $\\mathrm{Ext}$ groups are over $\\Z = \\Z_{(2)}$ unless marked otherwise. Denote by $\\underline{\\vb}$ a sequence of indecomposable elements $\\vb_i \\in \\pi_{(2^i-1)\\rho}^{C_2}B$. The aim of this section is to show that $\\Sigma^{2\\rho-4}\\Z^{B} \\simeq \\kappa_{M\\R}(\\underline{\\vb}; B)$. \n\nRecall that $\\kappa_{M\\R}(\\underline{\\vb}; B)$ is defined as follows: Given a sequence $\\underline{l} = (l_1,l_2,\\dots)$ with $l_i\\geq 0$, we denote by $B\/\\underline{\\vb}^{\\underline{l}}$ the spectrum $B\/(\\vb^{l_{i_1}}_{i_1}, \\vb^{l_{i_2}}_{i_2},\\dots)$, where $i_j$ runs over all indices such that $l_{i_j}> 0$. Set \n$$|\\underline{l}| = l_1|\\vb_1|+l_2|\\vb_2| + \\cdots $$\nThen \n$$\\kappa_{M\\R}(\\underline{\\vb}; B) = \\hcolim_{\\underline{l}} \\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\underline{\\vb}^{\\underline{l}},$$\nwhere $\\underline{l}$ runs over all sequences $\\underline{l}$ where all but finitely many $l_i$ are zero and $\\underline{1}$ denotes the constant sequence of ones. Furthermore, the $i$-th entry of $\\underline{l}-\\underline{1}$ is defined to be the maximum of $0$ and $l_i-1$. \n\nThus, to get a map $\\kappa_{M\\R}(\\underline{\\vb}; B) \\to \\Sigma^{2\\rho-4}\\Z^{B}$, we have to understand the homotopy classes of maps $B\/\\underline{\\vb}^{\\underline{l}} \\to \\Sigma^{2\\rho-4}\\Z^{B}$. This will be the content of the next subsection.\n\n\\subsection{Preparation}\nRecall the Mackey functor $\\underline{\\Z}^*$ defined by\n$$\\underline{\\Z}^*(\\Ctwo\/\\Ctwo) \\cong \\underline{\\Z}^*(\\Ctwo\/e)\\cong \\Z$$\nwith transfer equalling $1$ while restriction is multiplication by $2$. \n\\begin{lemma}\\label{lem:computation}\n As $\\underline{\\Z}[\\vb_1,\\vb_2,\\dots]$-modules, we have the following isomorphisms.\n \\begin{enumerate}\n \\item $\\underline{\\pi}^{C_2}_{*\\rho -4}B \\cong \\underline{\\Z}^*\\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]$ where $\\underline{\\Z}^*$ is generated by $1$ on underlying and by $2u^{-1}$ on $C_2$-equivariant homotopy groups.\n \\item $\\underline{\\pi}^{C_2}_{*\\rho -5}B = 0$\n \\item $\\pi^{C_2}_{*\\rho -6}B \\cong \\F_2\\{a^2\\vb_1(-1)\\} \\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]$. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n By Theorem \\ref{thm:BPR}, the groups $\\pi_{*\\rho -c}^{C_2}B$ are additively generated by nonzero elements of the form $x = a^l\\vb$ with $\\vb$ a monomial in the $\\vb_i(j)$. Let $\\vb_i(j)$ be the one occuring with minimal $i$, where $j$ is chosen such that $\\vb = \\vb_i(j)\\vb'$ with $\\vb'$ a monomial in the $\\vb_k$ (this is possible by the third relation in Theorem \\ref{thm:BPR}). Then $|x| = *\\rho +j2^{i+2}+l$ and $0 \\leq l< 2^{i+1}-1$. \n \n For $c=4$, this implies $j=-1$, $i=0$ and $l=0$. Thus, $x$ is of the form $\\vb_0(-1)\\vb'$. As the restriction of $\\vb_0(-1)$ to $\\pi_0^eB$ equals $2$, the result follows. \n \n For $c=5$, we must have $l \\geq 2^{i+2}-5$, which implies $l\\geq 2^{i+1}-1$ or $i=0$; in the latter case $l$ must be zero, which is not possible.\n \n For $c=6$, we must have $l = -j2^{i+2}-6$, which implies $l\\geq 2^{i+1}-1$ or $i\\leq 1$ and $j=-1$. As $i=0$ is again not possible, $x = a^2\\vb_1(-1)\\vb'$ with $\\vb' \\in \\pi_{*\\rho}^{C_2}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:AndersonQuot}\n For a sequence $\\ul = (l_1,l_2, \\dots)$, the map\n $$\\underline{\\pi}^{C_2}_{*\\rho+4} \\Z^{B\/\\uvb^{\\ul}} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul},\\Z) \\cong \\underline{\\Z} \\otimes_{\\Z} (\\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul})^*$$\n is an isomorphism, where $\\Z[\\vb_1,\\vb_2,\\dots]^{*} = \\mathrm{Hom}_\\Z(\\Z[\\vb_1,\\vb_2,\\dots], \\Z)$ (so that the gradings become nonpositive). Here, the second map is the dual of the map \n $$\\underline{\\Z}^* \\otimes_{\\Z} \\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul} \\to \\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul}$$\n sending $1 \\in \\underline{\\Z}^*(C_2\/C_2)$ to the image of $u^{-1}$ under the map $B\\to B\/\\uvb^{\\ul}$ and $1\\in\\underline{\\Z}^*(C_2\/e)$ to $1$. \n\\end{lemma}\n\\begin{proof}\n We have a short exact sequence\n $$0 \\to \\mathrm{Ext}(\\underline{\\pi}^{C_2}_{-*\\rho-5}B\/\\uvb^{\\ul}, \\Z) \\to \\underline{\\pi}^{C_2}_{*\\rho-4} \\Z^{B\/\\uvb^{\\ul}} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul}, \\Z) \\to 0.$$\n If $l_1=0$, then Corollary \\ref{Cor:QuotientBP} and Lemma \\ref{lem:computation} directly imply the statement. If $l_1\\neq 0$, Corollary \\ref{Cor:QuotientBP} only allows us to identify the homotopy Mackey functor in degree $-*\\rho-4$, but not the one in degree $-*\\rho-5$. We give a separate argument in this case.\n \n If $l_1\\neq 0$, consider the sequence $\\ul' = (0,l_2,l_3,\\dots)$ and the corresponding cofibre sequence\n $$ \\Sigma^{l_1\\rho}B\/\\uvb^{\\ul'} \\xrightarrow{\\vb_1^{l_1}} B\/\\uvb^{\\ul'} \\to B\/\\uvb^{\\ul} \\to \\Sigma^{l_1\\rho+1}B\/\\uvb^{\\ul'}.$$\n This induces a short exact sequence\n $$ 0 \\to (\\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul'})\/\\vb_1^{l_1} \\to \\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul} \\to \\{\\underline{\\pi}^{C_2}_{*\\rho-6}B\/\\uvb^{\\ul'}\\}_{\\vb_1^{l_1}} \\to 0.$$\n Here the last term denotes the sub Mackey functor of $\\underline{\\pi}^{C_2}_{*\\rho-6}B\/\\uvb^{\\ul'}$ killed by $\\vb_1^{l_1}$. By Corollary \\ref{Cor:QuotientBP} and Lemma \\ref{lem:computation}, we see that $\\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul} = 0$. \n \\end{proof}\n\nAs $B = BP\\R$ is not known to have an $E_\\infty$-structure, we have to work with $M\\R$-linear maps instead, for which the following lemma is useful:\n\n\\begin{lemma}\nThe map \n\\[\\Z^B \\simeq \\mathrm{Hom}_{M\\R}(M\\R, \\Z^B) \\to \\mathrm{Hom}_{M\\R}(B, \\Z^B)\\]\nis an equivalence.\n\\end{lemma}\n\\begin{proof}\nLet $e\\colon M\\R \\to M\\R$ be the Quillen--Araki idempotent. Recall that \n\\[B = \\hcolim \\left(M\\R \\xrightarrow{e} M\\R \\xrightarrow{e} \\cdots\\right).\\]\nThus, \n\\[\\Z^B \\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{e^*} \\Z^{M\\R} \\xrightarrow{e^*} \\Z^{M\\R}\\right).\\]\nHence,\n\\[\n\\mathrm{Hom}_{M\\R}(B, \\Z^B) \\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{e^*} \\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R}) \\xrightarrow{e^*} \\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R})\\right).\n\\]\nAs every $\\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R})$ is equivalent to a holim over $\\mathrm{Hom}_{M\\R}(M\\R, \\Z^{M\\R})\\simeq \\Z^{M\\R}$, connected by $e^*$, we get that \n$\\mathrm{Hom}_{M\\R}(B,\\Z^B)$ is the homotopy limit $\\hlim_{\\Z^-\\times \\Z^-}\\Z^{M\\R}$, where $\\Z^-$ denotes the poset of negative numbers and all connecting maps are $e^*$. This is equivalent to the homotopy limit indexed over the diagonal, which in turn is equivalent to the homotopy limit indexed over a vertical.\n\\end{proof}\n\nRecall that we want to show that $X = \\Sigma^{2\\rho-4}\\Z^B$ is equivalent to $\\kappa_{M\\R}(\\underline{\\vb},B)$. The reason for the choice of suspension is essentially (as before) that $H\\underline{\\Z} \\simeq \\Sigma^{2\\rho-4}H\\underline{\\Z}^*$.\n\n\\begin{prop}\\label{Cor:QuotientX}\nFor a sequence $\\ul = (l_1,l_2, \\dots)$, we have an isomorphism\n\\[\n \\underline{[\\Sigma^{*\\rho}B\/\\uvb^{\\ul}, X]}^{C_2}_{M\\R} \\cong \\underline{\\Z} \\otimes_{\\Z} (\\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul})^*,\n\\] \nnatural with respect to the maps $B\/\\uvb^{\\ul} \\to \\Sigma^{-|\\ul'-\\ul|\\rho}B\/\\uvb^{\\ul'}$ in the defining homotopy colimit for $\\kappa_{M\\R}(\\uvb; B)$ for $\\ul' = (l_1',l_2',\\dots)$ a sequence with $l_i' \\geq l_i$ for all $i\\geq 1$. \n\\end{prop}\n\\begin{proof}\nThe last lemma implies that we also have\n$$\\Z^{B\/\\uvb^{\\ul}} \\simeq \\mathrm{Hom}_{M\\R}(B\/\\uvb^{\\ul}, \\Z^B)$$\nas the functors $\\Z^{?}$ and $\\mathrm{Hom}_{M\\R}(?, \\Z^B)$ behave the same way with respect to cofibre sequences and (filtered) homotopy colimits. Then we just have to apply Lemma \\ref{lem:AndersonQuot}. \n\\end{proof}\n\n\\subsection{The theorem}\nWe first describe the homotopy groups of $X = \\Sigma^{2\\rho-4}\\Z^B$ with $B=BP\\R$ as before. \nBy Lemma \\ref{lem:AndersonQuot}, we get\n$$\\underline{\\pi}^{C_2}_{*\\rho}X \\cong \\mathrm{Hom}(\\underline{\\pi}_{(*+2)\\rho-4}^{C_2}B, \\Z) \\cong \\underline{\\Z} \\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]^{*}.$$ \n\nLet $\\underline{l}$ be a sequence with only finitely many nonzero entries. By Proposition \\ref{Cor:QuotientX}, the element $(\\underline{\\vb}^{\\underline{l}-\\underline{1}})^*$ induces a corresponding $M\\R$-linear map $\\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\vb^{\\underline{l}} \\to X$, which is unique up to homotopy. By this uniqueness, these maps are also compatible for comparable $\\underline{l}$. By Remark \\ref{rmk:hocolim}, this induces a map\n\\[\\kappa_{M\\R}(\\underline{\\vb},B) = \\hcolim_{\\underline{l}}\\left(\\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\vb^{\\underline{l}}\\right) \\;\\xrightarrow{h}\\, X,\\]\nwhere $\\underline{l}$ ranges over all sequences where only finitely many $l_i$ are nonzero.\n\n\\begin{thm}\\label{Thm:BPDuality}\nThis map $h\\colon \\kappa_{M\\R}(\\underline{\\vb}; B) \\to X$ is an equivalence of $\\Ctwo$-spectra. \n\\end{thm}\n\n\\begin{proof}\nBy Corollary \\ref{Cor:crucial}, we get on $\\underline{\\pi}_{*\\rho}$-level\n\\[\\clim_{\\underline{l}} \\Sigma^{-|\\underline{l}-\\underline{1}|}\\underline{\\Z}[\\vb_1,\\vb_2,\\dots]\/(\\vb_1^{l_1},\\dots) \\to \\underline{\\Z} \\otimes_\\Z \\Z[\\vb_1,\\dots]^*,\\]\nwhich is an isomorphism. The odd underlying homotopy groups of both sides are zero. To apply Lemma \\ref{lem:regrep}, it is left to show that $\\pi^{\\Ctwo}_{k\\rho-1}\\kappa_{M\\R}(\\underline{\\vb}; B) = 0$ for all $k\\in\\Z$. Again by Corollary \\ref{Cor:crucial}, it is even true that $\\pi^{\\Ctwo}_{k\\rho-1}(B\/\\vb^{\\underline{l}})$ is zero for all $k\\in\\Z$ and all sequences $\\underline{l}$.\n\\end{proof}\n\n\\section{Duality for regular quotients}\nThe goal of this section is to prove our main result Theorem \\ref{thm:main}:\n\\begin{thm}\\label{Thm:QuotientDuality}\nLet $(m_1,m_2,\\dots)$ be a sequence of nonnegative integers with only finitely many entries bigger than $1$. Denote by $\\underline{\\vb}'$ the sequence of $\\vb_i$ in $\\pi^{\\Ctwo}_{\\bigstar}M\\R$ such that $m_i = 0$ and by $m'$ the sum of all $(m_i-1)|\\vb_i|$ for $m_i> 1$. Then there is an equivalence\n$$\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-m'+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}'; B\/\\underline{\\vb}^{\\underline{m}}).$$\n\\end{thm}\nHere and for the rest of the section we will implicitly localize everything at $2$ again. Before we prove the theorem, we need some preparation.\n\n\\begin{lemma}\\label{Lem:AndersonQuotient}\nLet $\\underline{m} = (m_1,\\dots)$ be a sequence of nonnegative integers with a finite number $n$ of nonzero entries. Then \n\\[\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-|\\underline{m}|-n}(\\Z^B)\/\\underline{\\vb}^{\\underline{m}}.\\]\n\\end{lemma}\n\\begin{proof}\nLet $Y$ be an arbitrary ($\\Ctwo$-)spectrum and $\\Sigma^{|v|} Y \\xrightarrow{v} Y \\to Y\/v$ be a cofibre sequence. Then we have an induced cofibre sequence\n\\[\\Z^{Y\/v} \\to \\Z^Y \\xrightarrow{v} \\Sigma^{-|v|} \\Z^Y \\to \\Sigma \\Z^{Y\/v} \\simeq \\Sigma^{-|v|}(\\Z^{Y})\/v.\\]\nThus, $\\Z^{Y\/v} \\simeq \\Sigma^{-|v|-1}(\\Z^{Y})\/v$. The claim follows by induction. \n\\end{proof}\n\n\\begin{lemma}\\label{Lem:Nilpotence}\nThe element $\\vb_i^{3k}$ acts trivially on $B\/\\vb_i^k$ for every $i\\geq 1, k\\geq 1$.\n\\end{lemma}\n\\begin{proof}\nBy the commutativity of the diagram\n\\[\\xymatrix{\n\\Sigma^{k|\\vb_i|}B \\ar[d]^{\\vb_i^k}\\ar[r]& \\Sigma^{k|\\vb_i|}B\/\\vb_i^k \\ar[d]^{\\vb_i^k} \\ar[d]\\\\\nB \\ar[r] & B\/\\vb_i^k }\n\\]\nwe see that the composite $\\Sigma^{k|\\vb_i|} B \\to \\Sigma^{k|\\vb_i|} B\/\\vb_i^k \\xrightarrow{\\vb_i^k} B\/\\vb_i^k$ is zero, so that the latter map factors over an $M\\R$-linear map $\\Sigma^{2k|\\vb_i|+1}B \\to B\/\\vb_i^k$. As $[\\Sigma^{2k|\\vb_i|+1}B, B\/\\vb_i^k]_{M\\R}$ is a retract of $[\\Sigma^{2k|\\vb_i|+1}M\\R, B\/\\vb_i^k]_{M\\R} \\cong \\pi_{2k|\\vb_i|+1}^{C_2}B\/\\vb_i^k$, we just have to show that $\\vb_i^{2k}x = 0$ for every $x\\in \\pi_{2k|\\vb_i|+1}B\/\\vb_i^k$. \n\nWe have a short exact sequence\n\\[0 \\to (\\pi_\\bigstar^{\\Ctwo}B)\/\\vb_i^k \\to \\pi_\\bigstar^{\\Ctwo}(B\/\\vb_i^k)\\to \\left\\{\\pi^{\\Ctwo}_{\\bigstar -k|\\vb_i|-1} B\\right\\}_{\\vb_i^k} \\to 0.\\]\nAs $\\vb_i^k x$ clearly maps to zero, it is the image of a $y\\in (\\pi_\\bigstar^{\\Ctwo}B)\/\\vb_i^k$. But $\\vb_i^ky = 0$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:smash}\n We have $$B\/\\vb_i^l \\otimes_{M\\R} B\/\\vb_j^m \\simeq B\/(\\vb_i^l,\\vb_j^m).$$\n Furthermore, there is an equivalence\n \\[\\hcolim_l \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\simeq \\Sigma^{|\\vb_i| +1}B\/\\vb_i^m\\]\n of $M\\R$-modules if $m\\geq 1$.\n\\end{lemma}\n\\begin{proof}\nWe have $$B\\otimes_{M\\R} B \\simeq \\hcolim (B\\xrightarrow{e} B \\xrightarrow{e} \\cdots) \\simeq B,$$\nwhere $e$ denotes again the Quillen--Araki idempotent, \nand thus also\n$$B\/\\vb_i^l \\otimes_{M\\R} B\/\\vb_j^m \\simeq B\/(\\vb_i^l,\\vb_j^m).$$\n\nThus, the maps in the homotopy colimit in the lemma are induced by the following diagram of cofibre sequences:\n\\[\\xymatrix{\n\\Sigma^{|\\vb_i|}B\/\\vb_i^m \\ar[r]^-{\\vb_i^l}\\ar[d]^{\\mathrm{id}} & \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\ar[r]\\ar[d]^{\\vb_i} & \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\ar[d] \n\\\\\n\\Sigma^{|\\vb_i|}B\/\\vb_i^m \\ar[r]^-{\\vb_i^{l+1}} & \\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\ar[r]& \\Sigma^{-l|\\vb_i|}B\/\\vb_i^{l+1}\\otimes_{M\\R} B\/\\vb_i^m \n}\n\\]\nWe can assume that the homotopy colimit only runs over $l\\geq 3m$ so that by the last lemma the two cofibre sequences split and we get \n\\[\\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\simeq \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m.\\]\nThe corresponding map \n\\[\\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m \\to \\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m\\]\ninduces multiplication by $\\vb_i$ on the first summand, the identity on the second plus possibly a map from the second summand to the first. \n\nUsing this decomposition, it is easy to show that\n\\[\\hcolim_{l} \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m\\to \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m\\]\n(defined by the projection on the second summand for $l\\geq 3m$) is an equivalence. Indeed, on homotopy groups the map is clearly surjective. And if \n$$(x,y) \\in \\pi_\\bigstar^{C_2}\\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\oplus \\pi_{\\bigstar}^{C_2} \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m $$\nmaps to $0 \\in \\pi_{\\bigstar}^{C_2} \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m$, then $y = 0$ and $(x,0)$ represents $0$ in the colimit because $\\vb_i$ acts nilpotently. \n\\end{proof}\n\n\n\\begin{proof}[of theorem]\nAs in the theorem, let $\\underline{\\vb}'$ be the sequence of $\\vb_i$ such that $m_i = 0$ and also denote by $\\underline{\\vb}'' = (\\vb_{i_1},\\vb_{i_2},\\dots)$ the sequence of $\\vb_i$ such that $m_i \\neq 0$. \n\nWe begin with the case that $\\underline{m}$ has only finitely many\nnonzero entries (say $n$). By Lemma \\ref{Lem:AndersonQuotient} we see that \n\\[\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-|\\underline{m}|-n}(\\Z^B)\/\\underline{\\vb}^{\\underline{m}}.\\]\nCombining this with Theorem \\ref{Thm:BPDuality}, we obtain\n\\begin{align*}\n \\Z^{B\/\\underline{\\vb}^{\\underline{m}}} &\\simeq \\Sigma^{-|\\underline{m}|-n+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}, B)\/\\underline{\\vb}^{\\underline{m}} \\\\\n\t\t\t\t\t &\\simeq \\Sigma^{-|\\underline{m}|-n+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}',\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}))\n\\end{align*}\nThus, we have to show that $\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\Sigma^{|\\vb_{i_1}|+\\cdots |\\vb_{i_n}|+n}B\/\\underline{\\vb}^{\\underline{m}}$.\n\nBy Lemma \\ref{lem:smash}, we have an equivalence\n\\[(B\/\\underline{\\vb}^{\\underline{m}})\/(\\vb_{i_1}^{l_{i_1}}, \\dots, \\vb_{i_n}^{l_{i_n}}) \\simeq (B\/\\vb_1^{l_{i_1}}\\otimes_{M\\R} B\/\\vb_1^{m_{i_1}}) \\otimes_{M\\R}\\dots\\otimes_{M\\R} (B\/\\vb_n^{l_{i_n}}\\otimes_{M\\R} B\/\\vb_n^{m_{i_n}}).\\]\nIf we let now the homotopy colimit run over the sequences $(l_{i_1},\\dots, l_{i_n})$, we can do it separately for each tensor factor. Hence, we obtain again by Lemma \\ref{lem:smash} an equivalence\n\\[\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\Sigma^{|\\vb_{i_1}|+\\cdots |\\vb_{i_n}| +n} B\/\\underline{\\vb}^{\\underline{m}}.\\]\nThus, we have shown the theorem in the case that $\\underline{m}$ has only finitely many nonzero entries. \n\nWe prove the case that $\\underline{m}$ has possibly infinitely many nonzero entries by a colimit argument. Define $\\underline{m}_{\\leq k}$ to be the sequence obtained from $\\underline{m}$ by setting $m_{k+1}, m_{k+2}, \\dots$ to zero. Then $B\/\\underline{m} \\simeq \\hcolim_k B\/\\underline{m}_{\\leq k}$ and thus $\\Z^{B\/\\underline{m}} \\simeq \\hlim_k \\Z^{B\/\\underline{m}_{\\leq k}}$. Denote by $\\underline{\\vb}'_{\\leq k}$ the sequence of $\\vb_i$ such that $m_i = 0$ or $i>k$ and by $m'_k$ the quantity $|\\underline{m}_{\\leq k}-\\underline{1}|$; note that $m'_k = m'$ for $k$ large. \n\nWe have to show that the map\n\\[h\\colon \\Sigma^{-m'}\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}}) \\to \\hlim_k \\Sigma^{-m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\]\nis an equivalence. This map is defined as follows: We know that \n$$\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\hcolim_k \\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}}).$$\nUsing this, we get a map induced from the maps $\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}}) \\to \\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})$ for $k$ large.\n\nBy Corollary \\ref{Cor:crucial}, we can describe what happens on $\\pi_{*\\rho}^{C_2}$: The left hand side has as $\\Z$-basis monomials of the form $\\underline{\\vb}^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$. Likewise, \n$$\\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$$\nhas as $\\Z$-basis monomials of the form $\\uvb^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$ and $i\\leq k$. The maps in the homotopy limit induce the obvious inclusion maps. Thus, clearly the map \n$$\\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'}\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}})\\right) \\to \\lim_k \\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$$\nis an isomorphism. \n\nIt remains to show that $\\lim^1_k\\pi_{*\\rho+1}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$ vanishes. By Corollary \\ref{cor:rho+}, every term has as $\\F_2$-basis monomials of the form $a\\uvb^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$ and $i\\leq k$. The system becomes stationary in every degree, more precisely if $\\ast > -2^{k+1}$. Thus, the $\\lim^1$-term vanishes. A similar $\\lim^1$-argument also shows that the odd underlying homotopy groups of $\\hlim_k \\Sigma^{-m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})$ vanish.\n\nAs the source of $h$ is strongly even by Corollary \\ref{Cor:crucial} and by the arguments we just gave the morphism $h$ induces an isomorphism on $\\underline{\\pi}_{*\\rho}^{C_2}$ and on (odd) underlying homotopy groups, Lemma \\ref{lem:regrep} implies that $h$ is an equivalence. \n\\end{proof}\n\n\\vspace{1cm} \n\\part{Local cohomology computations}\\label{part:LocalCohomology}\n\nIn Part 4, we will describe the local cohomology spectral sequence in\nsome detail, and use it to understand the structure of the\n$H\\Zu$-cellularization of $BP\\R \\langle n \\rangle$. The calculation is not difficult,\nbut on the other hand it is quite hard to follow because it is made up\nof a large number of easy calculations which interact a little, and\nbecause one needs to find a helpful way to follow the $RO(\\Ctwo )$-graded\ncalculations. \n\nIn contrast the case of $k\\R$ is simple enough to be explained fully without\nfurther scaffolding, and it introduces many of the structures that we will want to\nhighlight. Since it may also be of wider interest than the general\ncase of $BP\\R \\langle n \\rangle$ we devote Section \\ref{sec:kRlcss} to it before\nreturning to the general case in Section \\ref{sec:BPRnlcss}. Section \\ref{sec:tmfotlcss} will then give a more detailed account in the interesting case $n=2$. \n\nLet us also recall some notation used throughout this part. As in the\nrest of the paper we work 2-locally, except when speaking about $k\\R$\nor $tmf_1(3)$ when fewer primes need be inverted. We often\nwrite $\\delta = 1-\\sigma \\in RO(C_2)$. We also recall the duality\nconventions from Section \\ref{sec:DAb}; in particular, for an\n$\\F_2$-vector space $V^{\\vee}$ equals the dual vector space\n$\\mathrm{Hom}_{\\F_2}(V,\\F_2)$ and for a torsionfree $\\Z$-module $M$, we set\n$M^*=\\mathrm{Hom}(M, \\Z)$. \n\n\nIf $R$ is a $C_2$-spectrum, we will use the notation $R^{C_2}_\\bigstar$ for its $RO(C_2)$-graded homotopy groups. We will also write $R^{hC_2}_{\\bigstar}=\\pi_{\\bigstar}^{C_2}(R^{(EC_2)_+})$ and similarly for geometric fixed points and the Tate construction. \n\n\\section{The local cohomology spectral sequence for $\\protect k\\R$}\n\\label{sec:kRlcss}\n\nThis section focuses entirely on the classical case of $k\\R$, where\nthere are already a number of features of interest. This gives a\nchance to introduce some of the structures we will use for the general\ncase. \n\n\\subsection{The local cohomology spectral sequence}\n\nGorenstein duality for $k\\R$ (Corollary \\ref{cor:kRGorD}) \nhas interesting implications for the coefficient ring, both\ncomputationally and structurally. \nWriting $\\bigstar$ for $RO(\\Ctwo)$-grading as usual, the local cohomology spectral\nsequence \\cite[Section 3]{G-M95} takes the following form. \n\n\\begin{prop} \n\\label{prop:kRlcss}\n There is a spectral sequence of $k\\R^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-4+\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{k\\R}).$$\nThe homotopy of the Anderson dual in an arbitrary degree $\\alpha \\in\nRO(C_2)$ lies in an exact sequence\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(k\\R^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}(\\Z^{k\\R}) \\longrightarrow \\mathrm{Hom}_{\\Z}(k\\R^{\\Ctwo}_{-\\alpha}, \\Z)\n\\longrightarrow 0. $$\nSince local cohomology is entirely in cohomological degrees 0 and 1,\nthe spectral sequence collapses to a short exact sequence \n$$0\\longrightarrow \\Sigma^{-1} H^1_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow \\Sigma^{-4+\\sigma}\n\\pi^{\\Ctwo}_{\\bigstar}(\\Z^{k\\R}) \\longrightarrow H^0_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow\n0. $$\nThis sequence is not split, even as abelian groups. \n\\end{prop}\n\nOne should not view Proposition \\ref{prop:kRlcss} as an algebraic\nformality: it embodies the fact that $k\\R^{\\Ctwo}_{\\bigstar}$ is a very special\nring. To illustrate this, we recall the calculation of \n$k\\R^{\\Ctwo}_{\\bigstar}$ in Subsection \\ref{sec:kRgroups}. In Subsection\n\\ref{subsec:kRloccoh} we calculate its local cohomology, and how\nthe Gorenstein duality isomorphism with the known homotopy of the Anderson \ndual works. \n\n\\subsection{The ring $\\protect k\\R^{\\Ctwo}_{\\bigstar}$}\\label{sec:kRgroups}\n\nOne may easily calculate $k\\R^{\\Ctwo}_{\\bigstar}$. This has already been done in \\cite{B-G10}, but we sketch a slightly different method. We will first calculate $k\\R^{h\\Ctwo}_{\\bigstar} $ and then use the Tate square \\cite{GMTate}. \n\nIn the homotopy fixed point spectral sequence\n$$\\Z[\\vb, a, u^{\\pm 1}]\/2a \\Rightarrow k\\R^{hC_2}_{\\bigstar}$$\nall differentials are generated by $d_3(u)=\\vb a^3$. Indeed, this differential is forced by $\\eta^4 = 0$ and there is no room for further ones. \nIt follows that $U=u^2$ is an infinite cycle, and so the\nwhole ring is $U$-periodic: \n$$k\\R^{h\\Ctwo}_{\\bigstar}=BB [U,U^{-1}], $$\nwhere $BB$ is a certain `basic block'. This basic block is a sum\n$$BB=BR\\oplus (2u)\\cdot \\Z [\\vb]$$\nas $BR$-modules, where \n$$ BR=\\Z [\\vb,a]\/(2a, \\vb a^3).$$\n\nIt is worth illustrating $BB$ in the plane (with $BB_{a+b\\sigma}$\nplaced at the point $(a,b)$). The squares and circles represent copies of $\\Z$, and\nthe dots represent copies of $\\mathbb{F}_2$. The left hand vertical column\nconsists of 1 (at the origin, $(0,0)$) and the powers of $a$, but the feature to concentrate on\nis the diagonal lines representing $\\Z [\\vb]$ submodules. These are\neither copies of $\\Z [\\vb]$ or of $\\mathbb{F}_2 [\\vb]$ or simply copies of $\\mathbb{F}_2$. \n\n$$\\begin{tikzpicture}[scale =1]\n\\draw[step=0.5, gray, very thin] (-3,-3) grid (3, 3);\n\\draw (-1.5,1.5 ) node[anchor=east, draw=orange]{\\Large{BB}};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (0,-\\y\/2) node[anchor=east] {$a^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (0,-\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw]\n{};\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2-1\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\draw [->] (0,0)-- (3,3);\n\\node at (0,0) [shape = rectangle, draw]{};\n\\draw (0,0) node[anchor=east]{1};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (\\y\/2,\\y\/2) node[anchor=east] {$\\vb^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (\\y\/2,\\y\/2) [shape=rectangle, draw] {};\n\n\\draw[red] (0,-0.5)--(3,2.5);\n\\draw[red] (0,-1.0)--(3,2.0);\n\n\\draw [->](1,-1)-- (3,1);\n\\node at (1,-1) [shape=circle, draw] {};\n\\draw (1,-1) node[anchor=east]{$2u$};\n\n\\foreach \\y in {1,2,3,4}\n\\node at (1+\\y\/2,-1+\\y\/2) [shape=circle, draw] {};\n\\end{tikzpicture}$$\n\nProceeding with the calculation, we may invert $a$ to find the\nhomotopy of the Tate spectrum $k\\R^t=F(E(C_2)_+, k\\R ) \\wedge S^{\\infty\n \\sigma}$: \n$$k\\R^{t\\Ctwo}_{\\bigstar}=\\mathbb{F}_2 [a,a^{-1}][U,U^{-1}]. $$\nOne also sees that the homotopy of the geometric fixed points (the\nequivariant homotopy of $k\\R^{\\Phi}=k\\R \\wedge S^{\\infty\\sigma}$) is \n$$k\\R^{\\Phi \\Ctwo}_{\\bigstar}=\\mathbb{F}_2 [a,a^{-1}][U] $$\nusing the following lemma:\n\\begin{lemma}\n\\label{lem:Phiconn}\n Let $X$ be a $\\Ctwo$-spectrum which is non-equivariantly connective\n and such that $X^{\\Ctwo} \\to X^{h\\Ctwo}$ is a connective cover. Then $X^{\\Phi \\Ctwo} \\to X^{t\\Ctwo}$ is a connective cover as well.\n\\end{lemma}\n\n\n\\begin{proof}\n This follows from the diagram of long exact sequences\n \\[\\xymatrix{\n \\pi_kX_{h\\Ctwo} \\ar[r]\\ar[d] & \\pi_kX^{\\Ctwo} \\ar[r]\\ar[d]& \\pi_kX^{\\Phi \\Ctwo} \\ar[r]\\ar[d] & \\pi_{k-1}X_{h\\Ctwo} \\ar[r]\\ar[d] & \\pi_{k-1}X^{\\Ctwo}\\ar[d] \\\\\n \\pi_kX_{h\\Ctwo} \\ar[r] & \\pi_kX^{h\\Ctwo} \\ar[r]& \\pi_kX^{t\\Ctwo} \\ar[r] & \\pi_{k-1}X_{h\\Ctwo} \\ar[r] & \\pi_{k-1}X^{h\\Ctwo},\n }\n \\]\n the fact that $X_{h\\Ctwo}$ is connective and the $5$-lemma. \n\\end{proof}\n\n\nNow the Tate square \n$$\\diagram \nk\\R \\rto \\dto & k\\R \\wedge S^{\\infty \\sigma} \\dto\\\\\nk\\R^{(E\\Ctwo)_+} \\rto & k\\R^{(E\\Ctwo)_+} \\wedge S^{\\infty \\sigma} \n\\enddiagram$$\ngives $k\\R^{\\Ctwo}_{\\bigstar}$.\n\nIt is convenient to observe that the two\nrows are of the form $M\\longrightarrow M[1\/a]$, so that the fibre is $\\Gamma_a\nM$. Since the two rows have equivalent fibres, we calculate the\nhomotopy of the second and obtain \n$$k\\R^{\\bigstar}_{hC_2}=NB [U, U^{-1}], $$\nwhere $NB$ is quickly calculated as the $(a)$-local cohomology\n$H^*_{(a)}(BB)$ (and named $NB$ for `negative block'). The element\n$a$ acts vertically and we can immediately read off the answer: the\ntower $\\Z [a]\/(2a)$ gives some $H^1$, and the rest is $a$-power torsion:\n$$NB=BB'\\oplus \\Sigma^{-\\pp} \\mathbb{F}_2 [a]^{\\vee}, $$\nwhere $BB' \\subset BB$ is the sub-$BR$-module\ngenerated by $2, \\vb, 2u$ (Informally, we may say that $BB'$ omits from $BB$ all monomials $a^k$ for\n$k\\geq 1$ and the generator 1). \nNote that $NB$ is placed so that its element $2$ is in degree 0 for\nease of comparison to $BB$; all occurrences of $NB$ in $k\\R^{\\Ctwo}_{\\bigstar}$\ninvolve nontrivial suspensions. \n\n\n\n\n\nAgain, it is helpful to display the negative block. This differs from\n$BB$ in that the powers of $a$ have been deleted, and replaced by a\nnew left hand column $\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$. The other new\nfeature is that the copy of $\\Z [\\vb]$ generated by $1$ has been\nreplaced by the kernel $(2,\\vb)$ of $\\Z\n[\\vb]\\longrightarrow \\mathbb{F}_2$, as indicated by the circle at the origin, labelled by its generator 2.\n\n$$\\begin{tikzpicture}[scale =1]\n\\draw[step=0.5, gray, very thin] (-3,-3) grid (3, 3);\n\\draw (-1.5,1.5 ) node[anchor=east, draw=orange]{\\Large{NB}};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (-0.5,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\\draw[red] (-0.5,0.5)--(-0.5,3);\n\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw]\n{};\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2-1\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\n\\draw [->] (0,0)-- (3,3);\n\\node at (0,0) [shape = circle, draw]{};\n\\draw (0,0) node[anchor=east]{2};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (\\y\/2,\\y\/2) node[anchor=east] {$\\vb^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (\\y\/2,\\y\/2) [shape=rectangle, draw] {};\n\\draw[red] (0.5,0)--(3,2.5);\n\\node at (0.5,0) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\\draw[red] (0.5,-0.5)--(3,2.0);\n\\node at (0.5,-0.5) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\draw [->](1,-1)-- (3,1);\n\\node at (1,-1) [shape=circle, draw] {};\n\\draw (1,-1) node[anchor=east]{$2u$};\n\n\\foreach \\y in {1,2,3,4}\n\\node at (1+\\y\/2,-1+\\y\/2) [shape=circle, draw] {};\n\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw[red] (\\y\/2,\\y\/2) -- (\\y\/2, \\y\/2-1);\n\\end{tikzpicture}$$\n\\vspace{0.2cm}\n\nThe Tate square then lets us read off \n$$k\\R^{\\Ctwo}_{\\bigstar}=\\bigoplus_{k\\leq -1} NB\\cdot \\{ U^{k}\\} \\oplus\n\\bigoplus_{k\\geq 0} BB\\cdot \\{U^{k}\\} =(U^{-1} \\cdot NB [U^{-1}] )\\oplus BB[U] $$\nThe $\\Z [U]$ module structure is given by letting $U$ act in the\nobvious way on the $NB$ and $BB$ parts, and by the maps\n$$NB \\longrightarrow BB'\\longrightarrow BB $$\nin passage from the $U^{-1}$ factor of $NB$ to the $U^0$ factor of $BB$.\n\nPerhaps it is helpful to note that with the exception of the towers\n$U^{-k}\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$, we have a subring of\n$BB[U,U^{-1}]$, which consists of blocks $BB\\cdot U^i$ for \n$i \\geq 0$ and blocks $BB'\\cdot U^i$ for $i<0$.\n\n\\subsection{Local cohomology}\n\\label{subsec:kRloccoh}\nRecall that we are calculating local cohomology with respect to the principal\nideal $(\\vb)$ so that we only need to consider $k\\R^{\\Ctwo}_{\\bigstar}$ as a\n$\\Z [\\vb]$-module. As such it is a sum of suspensions of the blocks\n$BB$ and $NB$, so we just need to calculate the local cohomology of\nthese. \n\nMore significantly, $\\Z [\\vb]$ is graded over multiples of the regular\nrepresentation, so local cohomology calculations may be performed on one diagonal\nat a time (i.e., we fix $n$ and consider gradings $n+*\\rho$). The only modules that occur are\n$$\\Z [\\vb], \\mathbb{F}_2 [\\vb], \\mathbb{F}_2 \\mbox{ and the ideal } (2,\n\\vb)\\subseteq \\Z[\\vb], $$\neach of which has local cohomology that is very easily calculated. \n\n\n\n\n\\begin{lemma}\nThe local cohomology of the basic block $BB$ is as follows. \n$$H^0_{(\\vb)}(BB)=a^3 \\mathbb{F}_2 [a]$$\n$$H^1_{(\\vb)}(BB)=\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\Sigma^{-\\rho+2\\delta} \\Z [\\vb]^{*}\\oplus\n\\Sigma^{-\\rho-\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}\\oplus \\Sigma^{-\\rho-2\\sigma}\n\\mathbb{F}_2 [\\vb]^{\\vee}. $$\n\\end{lemma}\n\n\\begin{proof}\nThe local cohomology is the cohomology of the complex\n$$BB\\longrightarrow BB[1\/\\vb]. $$ \nIt is clear that \n$$BB[1\/\\vb]=\\Z [\\vb,\\vb^{-1}]\\oplus u\\cdot \\Z [\\vb,\\vb^{-1}]\\oplus \na\\cdot \\mathbb{F}_2 [\\vb,\\vb^{-1}]\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb,\\vb^{-1}]\\qedhere$$\n\\end{proof}\n\n\nTurning to $NB$, we recall that $NB=BB'\\oplus \\Sigma^{-\\delta} \\mathbb{F}_2\n[a]^{\\vee}$, and we have a short exact sequence\n$$0\\longrightarrow BB'\\longrightarrow BB\\longrightarrow \\mathbb{F}_2 [a]\\longrightarrow 0. $$ \nThe local cohomology is thus easily deduced from that of $BB$.\n\n\n\\begin{lemma}\nThe local cohomology of the negative block $NB$ is as follows. \n$$H^0_{(\\vb)}(NB)=\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$$\n$$H^1_{(\\vb)}(NB)=\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\mathbb{F}_2 \\oplus \\Sigma^{-\\rho+2\\delta} \\Z [\\vb]^{*}\\oplus\n\\Sigma^{-\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}\\oplus \\Sigma^{-2\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}$$\nMore properly, the $\\Z [\\vb]$-module structure of the sum of the first\ntwo terms is \n$$\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\mathbb{F}_2 \\cong \\Z [\\vb]^*\/(2\\cdot 1^*).$$\n\\end{lemma}\n\n\\begin{proof}\nThe local cohomology is the cohomology of the complex\n$$NB\\longrightarrow NB[1\/\\vb]. $$\n\nIt is clear that $NB[1\/\\vb]=BB[1\/\\vb]$, which makes the part coming\nfrom the $2$-torsion clear. For the $\\Z$-torsion free part, it is\nhelpful to consider the exact sequence\n$$0\\longrightarrow (2, \\vb) \\longrightarrow \\Z [\\vb]\\longrightarrow \\mathbb{F}_2 \\longrightarrow 0$$\nand then consider the long exact sequence in local cohomology.\n\\end{proof}\n\nImmediately from the defining cofibre sequence $\\Gamma_{\\vb}k\\R \\longrightarrow\nk\\R \\longrightarrow k\\R [1\/\\vb]$ we see that there is a short exact sequence\n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow\n\\pi^{\\Ctwo}_{\\bigstar}(\\Gamma_{(\\vb)}k\\R)\\longrightarrow\nH^0_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow 0. $$\nThis gives $\\pi^{\\Ctwo}_{\\bigstar}(\\Gamma_{(\\vb)}k\\R)$ up to\nextension. The Gorenstein duality isomorphism can be used to resolve the remaining extension\nissues, and the answer is recorded in the proposition below. \n\nThe diagram Figure \\ref{fig:GBBkR} should help the reader interpret the statement and proof of\nthe calculation of the homotopy of $\\Gamma_{(\\vb)}k\\R$. We have\nomitted dots, circles and boxes except at the ends of diagonals or\nwhere an additional generator is required. The vertical lines denote\nmultiplication by $a$ and the dashed vertical line is an exotic\nmultiplication by $a$ that is not visible on the level of local\ncohomology. The green diamond does not denote a class, but marks the\npoint one has to reflect (non-torsion classes) at to see Anderson\nduality. Torsion classes are shifted by $-1$ after reflection (i.e., shifted\none step horizontally to the left). \n\n\\begin{center}\n\\begin{figure}\\includegraphics{GBBkR}\n\\caption{Gorenstein duality for $k\\R$ \\label{fig:GBBkR}}\n\\end{figure}\n\\end{center}\n\n\n\\begin{prop}\nThe homotopy of the derived $\\vb$-power torsion is given by \n$$\\pi_{\\bigstar}^{\\Ctwo} (\\Gamma_{(\\vb)}k\\R)\\cong (U^{-1}\\cdot GNB [U^{-1}]) \\oplus GBB [U]$$\nwhere $GBB$ and $GNB$ are based on the local cohomology of $BB$ and\n$NB$ respectively, and described as follows. We have \n$$GBB= \\Sigma^{-2-\\sigma} \\left[ \n\\Z [\\vb]^{*}\\oplus a\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus u\\cdot N \\right] $$\nwhere $N$ (with top in degree 0) is given by an exact sequence\n$$0\\longrightarrow \\Z [\\vb]^* \\longrightarrow N \\longrightarrow \\mathbb{F}_2 [a]\\longrightarrow 0, $$\nnon-split in degree 0.\n\nSimilarly, \n$$GNB= \\Sigma^{-1} \n\\left[ \\Z [\\vb]^{*}\/(2 \\cdot (1^*) ) \\oplus \na\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus \\Sigma^{1-3\\sigma}\\Z [\\vb]^* \\oplus \\Sigma^{\\sigma}\\mathbb{F}_2\n[a]^{\\vee} \\right] $$\nwhere the action of $a$ is as suggested by the sum decomposition\nexcept that multiplication by $a$ is non-trivial wherever possible\n(i.e., when one dot is vertically above another, or where a box is\nvertically above a dot). \n\\end{prop}\n\n\\begin{proof}\nWe first note that the contributions from the different blocks do not\ninteract. Indeed, the only time that different blocks give\ncontributions in the same degree come from the $\\mathbb{F}_2 [a]$ towers of\n$BB$: one class in that degree is $\\vb$-divisible (and not killed\nby $\\vb$) and the other class is annihilated by $\\vb$. We may therefore \nconsider the blocks entirely separately.\n\nThe block $GBB$ comes from the local cohomology of $BB$ and therefore\nlives in a short exact sequence \n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}BB) \\longrightarrow\nGBB \\longrightarrow H^0_{(\\vb)}(BB) \\longrightarrow 0$$\n\nThe block $GNB$ comes from the local cohomology of $NB$\nand therefore lives in a short exact sequence \n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}NB) \\longrightarrow\nGNB \\longrightarrow H^0_{(\\vb)}(NB) \\longrightarrow 0$$\n\nMost questions about module structure over $BB[U]$ are resolved by\ndegree, but there are two which remain. These can be resolved Gorenstein duality \\ref{cor:kRGorD} and the known module structure in $\\Z^{k\\R}$. \n\nIn $GBB$, the additive\nextension in $\\pi^{\\Ctwo}_{-3\\sigma}$ is non-trivial: \n$$\\pi_{-3\\sigma}^{\\Ctwo} (\\Gamma_{(\\vb)}k\\R)\\cong \\Z. $$\nAlso the multiplication by $a$ \n$$\\mathbb{F}_2 \\cong GNB_{-1+\\sigma} \\to GNB_{-1}\\cong \\mathbb{F}_2$$\nis nonzero (where $GNB_{-1+\\sigma}$ corresponds to $\\pi^{C_2}_{-5+5\\sigma}(\\Gamma_{(\\vb)}k\\R)$ in the $U^{-1}$-shift). \n\\end{proof}\n\n\\begin{remark}\n It is striking that the duality relates the top $BB$ to the bottom\n$NB$ (i.e., Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of $BB$ to $NB$), and it takes the\nbottom $NB$ to the top $BB$ (i.e., Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of $NB$ to $BB$). \nIndeed, as commented after Lemma \\ref{lem:Phiconn}, since $NB=\\Gamma_{(a)}BB$, we have\n$$\\Sigma^{2+\\sigma}\\Gamma_{(\\vb)}BB\\simeq (\\Gamma_{(a)} BB)^*$$\nand \n$$\\Gamma_{(\\vb, a)} BB\\simeq \\Sigma^{-2-\\sigma} BB^*, $$\nwith the second stating that $BB$ is Gorenstein of shift $-2-\\sigma$\nfor the ideal $(a, \\vb )$. \n\n\nBy extension, Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of all copies of $BB$ to all copies\nof $NB$ and vice versa. This might suggest separating $k\\R$ into a\npart with homotopy $BB[U]$, giving a cofibre sequence \n$$\\langle BB[U]\\rangle\\longrightarrow k\\R \\longrightarrow \\langle U^{-1}NB[U^{-1}]\\rangle , $$\nwhere the angle brackets refer to a spectrum with the indicated\nhomotopy. However one may see that there is no $C_2$-spectrum with homotopy the Mackey functor corresponding to $BB[U]$\n(considering the $b\\sigma$ and $(b+1)\\sigma$ rows one sees that the\nnon-equivariant homotopy of the spectrum would be zero up to about \ndegree $2b$; taking all rows together it would have to be\nnon-equivariantly contractible and hence $a$-periodic). Similarly,\nthere is no spectrum with homotopy $U^{-1}NB[U^{-1}]$, so these dualities are purely\nalgebraic. \n\\end{remark}\n\n\n\n\\section{The local cohomology spectral sequence for $\\protect BP\\R \\langle n \\rangle$}\n\\label{sec:BPRnlcss}\n\n\n\nGorenstein duality for $BP\\R \\langle n \\rangle$ (Example \\ref{ex:BPRn}) \nhas interesting implications for the coefficient ring, both\ncomputationally and structurally. \nWriting $\\bigstar$ for $RO(\\Ctwo)$-grading as usual, the local cohomology spectral\nsequence \\cite[Section 3]{G-M95} takes the form described in the\nfollowing proposition.\n We now revert to our standard assumption of\nworking 2-locally, so that $\\Z$ means the 2-local integers. \n\n\n\\begin{prop} \n\\label{prop:BPRnlcss}\nThere is a spectral sequence of $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{\\Jb_n}(BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{BP\\R \\langle n \\rangle})$$\nfor $\\Jb_n = (\\vb_1,\\dots, \\vb_n)$. \nThe homotopy of the Anderson dual in an arbitrary degree $\\alpha \\in\nRO(C_2)$ is easily calculated\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(BP\\R \\langle n \\rangle^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}\\Z^{BP\\R \\langle n \\rangle} \\longrightarrow \\mathrm{Hom}_{\\Z}(BP\\R \\langle n \\rangle^{\\Ctwo}_{-\\alpha}, \\Z) \\longrightarrow 0. $$\nFor $n\\geq 2$ the local cohomology spectral sequence has some non-trivial\ndifferentials. \n\\end{prop}\n\nOne should not view Proposition \\ref{prop:BPRnlcss} as an algebraic\nformality: it embodies the fact that $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$ is a very special\nring. \n\nIn the present section we will discuss the implications of this for\nthe coefficient ring for general $n$. The perspective is a bit distant\nso the reader is encouraged to refer back to $k\\R$ (i.e., the case\n$n=1$) in Section \\ref{sec:kRlcss} to anchor the generalities. \n\nHowever the case $n=1$ is too simple to show some of what happens, so \nwe will also illustrate the case $tmf_1(3)$ (i.e.,\nthe case $n=2$) in Section \\ref{sec:tmfotlcss}.\n\n\\subsection{Reduction to diagonals}\nFor brevity we write $R_{\\bigstar}=BP\\R \\langle n \\rangle_{\\bigstar}^{\\Ctwo}$. Because the ideal\n$\\Jb_n=(\\vbn{1}, \\ldots , \\vbn{n})$\n is generated by elements whose degrees are a multiple of\n$\\rho$, we can\ndo $\\Jb_n$-local cohomology calculations over the subring $R_{*\\rho}$\nof elements in degrees which are multiples of $\\rho$. \n\nThus, for an $R_{\\bigstar}$-module $M_{\\bigstar}$ we have a direct sum decomposition\n$$M_{\\bigstar}=\\bigoplus_{d} M_{d+*\\rho}$$\nas $R_{*\\rho}$-modules, where we refer to the gradings $d+*\\rho$ as the\n{\\em $d$-diagonal}. Hence, we also have\n$$H^i_{\\Jb_n}(M_{\\bigstar})=\\bigoplus_d H^i_{\\Jb_n}(M_{d+*\\rho}). $$\n(We have abused notation by also writing $\\overline{J}_n$ for the ideal of\n$R_{*\\rho}$ generated by $\\vbn{1}, \\ldots , \\vbn{n}$.)\n\n\\subsection{The general shape of $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$}\nBy the description at the end of Section \\ref{sec:BPRnC2}, we have an isomorphism\n$$R_{\\bigstar}= U^{-1}\\cdot NB[U^{-1}] \\oplus BB [U]$$\nwith $BB$ and $NB$ as described there. It is easy to see that $BB$ and $NB$ decompose as $R_{*\\rho}$-modules into modules of a certain form we will describe now. We will implicitly $2$-localize everywhere. \n\nThe modules $BB$ and $NB$ decompose into are\n$$P=R_{*\\rho}=\\Z[\\vbn{1}, \\ldots, \\vbn{n}] \\mbox{ and } \\Pb{s}=P\/(\\vbn{0}, \\ldots,\n\\vbn{s})=\\mathbb{F}_2 [\\vbn{s+1},\\ldots , \\vbn{n}] $$\nfor $s\\geq 0$\nand the ideals expressed by the exact sequences \n\\begin{align*} 0\\longrightarrow (2, \\vbn{1}, \\ldots , \\vbn{t})\\longrightarrow P \\longrightarrow\n \\Pb{t}\\longrightarrow 0 \\end{align*}\nor \n\\begin{align*}0\\longrightarrow (\\vbn{s+1}, \\ldots , \\vbn{t})\\longrightarrow \\Pb{s}\\longrightarrow \\Pb{t}\\longrightarrow 0\\end{align*}\nwith $s\\geq 0$.\n\nTheir local cohomology is easily calculated. In the first two cases, the modules only have local cohomology in a\nsingle degree\n\\begin{align*}H_{\\Jb_n}^*(P)&=H_{\\Jb_n}^n(P)=P^*(-D_n\\rho) \\\\\nH_{\\Jb_n}^*(\\Pb{s})&=H_{\\Jb_n}^{n-s}(\\Pb{s})=\\Pb{s}^{\\vee}((D_s-D_n)\\rho). \\end{align*}\nThe top non-zero degree of $P^*$ is zero, so that $1^* \\in\nP^*(-D_n\\rho)$ is in degree $-D_n\\rho = -|\\vb_1|-\\cdots -\n|\\vb_n|$. We alert the reader to the fact that star is used in two ways: occasionally in\n$H^*$ to mean cohomological grading and rather frequently here in\n$P^*$ to mean the $\\Z$-dual of $P$. \n\nNow we turn to the ideal $(\\vbn{s+1}, \\ldots , \\vbn{t})$. If $t=s+1$ the ideal is principal\nand $(\\vbn{s+1})\\cong \\Pb{s}((s+1)\\rho)$; thus we get a single local cohomology group\n$$H^{n-s}_{\\Jb_n} ((\\vbn{s+1}) \\Pb{s})=\\Pb{s}^{\\vee}((D_s-D_n+s+1)\\rho)$$\nas can be seen from the long exact sequence of local cohomology. \n\nOtherwise we get two local cohomology groups\n$$H^{n-s}_{\\Jb_n} ((\\vbn{s+1}, \\ldots , \\vbn{t})\\Pb{s})=\\Pb{s}^{\\vee}((D_n-D_s)\\rho)\n\\mbox{ and } H^{n-t+1}_{\\Jb_n} ((\\vbn{s+1}, \\ldots , \\vbn{t})\n\\Pb{s})=\\Pb{t}^{\\vee}((D_n-D_t)\\rho).$$\n\nThe case of $(2, \\vbn{1}, \\ldots , \\vbn{t})$ is similar but with an extra case. The case $t=0$ is easy since then\n$(2)\\cong P$ so the local cohomology is all in cohomological degree $n$ where it is\n$P^*(-D_n\\rho)$. If $t=1$ we again get a single local cohomology group\n$$H^{n}_{\\Jb_n} ((2,\\vbn{1}) P)=P^*(-D_n\\rho)\\oplus \\Pb{1}^{\\vee}((D_{1}-D_n)\\rho).$$\nOtherwise we get two local cohomology groups\n$$H^{n}_{\\Jb_n} ((2, \\ldots , \\vbn{t})P)=P^*(-D_n\\rho)\n\\mbox{ and } H^{n-t+1}_{\\Jb_n} ((2, \\ldots , \\vbn{t})P) =\\Pb{t}^{\\vee}((D_t-D_n)\\rho).$$\n\n\n\\subsection{The special case $n=1$}\nThe best way to make the patterns apparent is to look at the simplest\ncases. In this section we begin with $k\\R^{C_2}_{\\bigstar}$ as treated in\nSection \\ref{sec:kRlcss} above, and we encourage the reader to relate\nthe calculations here to the diagrams in Section \\ref{sec:kRlcss}. In that case, \n$$P=k\\R^{C_2}_{*\\rho}=\\Z [\\vbn{1}], \\Pb{0}=\\mathbb{F}_2 [\\vbn{1}] \\mbox{ and } \\Pb{1}=\\mathbb{F}_2.$$\n\nDisplaying $BB$ by $d$-diagonal, we have\n\n$$\\begin{array}{c|cc}\n&BB&(n=1)\\\\\n\\hline\nd&1&u\\\\\n\\hline\n0&P&\\\\\n1&\\Pb{0}&\\\\\n2&\\Pb{0}&\\\\\n3&\\Pb{1}&\\\\\n4&\\Pb{1}&(2)P\\\\\n5&\\Pb{1}&\\\\\n6&\\Pb{1}&\\\\\n7&\\Pb{1}&\\\\\n8&\\Pb{1}&\\\\\n\\end{array}$$\n\\vspace{0.2cm}\n\nThe position of the modules along the $d$-diagonal can be inferred\nfrom the label at the top of the column. Thus the first column has\ngenerators in degree $-d\\sigma$, and the second column similarly, but\n in the column of $u$ (namely the 2-column). Noting that $u$ is on the\n 4-diagonal, the $d$th row has generators in $|u| -(d-4)\\sigma = 2-(d-2)\\sigma$. For example, along the 4-diagonal we have $a^4\\Pb{1}\n\\oplus (2u)P$.\n\n\n\nTaking local cohomology, and shifting $H^s_{\\Jb_n}$ down by $s$ (as in the local cohomology spectral sequence), we have\n$$\\begin{array}{c|cc}\n&H^*_{(\\vb_1)}(BB)&(n=1)\\\\\n\\hline\nd&1&u\\\\\n\\hline\n-1&{\\color{brown}P^*(-2\\rho)}&\\\\\n0&{\\color{brown} \\Pb{0}^{\\vee}(-2\\rho)}&\\\\\n1&{\\color{brown}\\Pb{0}^{\\vee}(-2\\rho)}&\\\\\n2&&\\\\\n3&\\Pb{1}&{\\color{brown}P^*(-2\\rho)}\\\\\n4&\\Pb{1}&\\\\\n5&\\Pb{1}&\\\\\n6&\\Pb{1}&\\\\\n7&\\Pb{1}&\\\\\n8&\\Pb{1}&\\\\\n\\end{array}$$\n\\vspace{0.2cm}\n\nHere, we colored $H^1$-groups brown. Note that shifting down by $s$ both lowers $d$ by $s$ and adds a shift by $-s\\rho$. For example, considering the 3-diagonal of this table, the $\\Pb{1}$ comes directly\nfrom the 3-diagonal of $BB$, whilst the $P^*(-2\\rho)$ comes from the\n$(2)P$ on the 4-diagonal of $BB$; the local cohomology is\n$P^*(-\\rho)$, but its diagonal is shifted by $-1$ since it is a first\nlocal cohomology, and because it is by reference to the 2-column the\nshift is $-\\rho$. The top of this module is calculated by\nreference to the column of $|u|$ (i.e., the 2-column), and has top in degree\n $2-(3-2)\\sigma-2\\rho=-3\\sigma$.\n\nWe saw in Section \\ref{sec:kRlcss} that the two modules on the\n3-diagonal give a non-trivial additive extension (in degree\n$-3\\sigma$) after running the spectral sequence. \n\n\n\\subsection{The special case $n=2$}\n\\label{subsec:tmfotloccoh}\nContinuing our effort to make patterns visible, we consider\n$tmf_1(3)^{C_2}_{\\bigstar}$ in this subsection (i.e., the case $n=2$). With\n$\\Z$ denoting the integers with 3 inverted here, this has\n$$P=tmf_1(3)^{C_2}_{*\\rho}=\\Z [\\vbn{1}, \\vbn{2}], \\Pb{0}=\\mathbb{F}_2 [\\vbn{1},\n\\vbn{2}], \\Pb{1}=\\mathbb{F}_2 [\\vbn{2}] \\mbox{ and } \\Pb{2}=\\mathbb{F}_2.$$\n\nThus for $n=2$ we have\n$$\\begin{array}{c|cccc}\n&BB&(n=2)&&\\\\\n\\hline\nd&1&u&u^2&u^3\\\\\n\\hline\n0&P&&&\\\\\n1&\\Pb{0}&&&\\\\\n2&\\Pb{0}&&&\\\\\n3&\\Pb{1}&&&\\\\\n4&\\Pb{1}&(2)&&\\\\\n5&\\Pb{1}&&&\\\\\n6&\\Pb{1}&&&\\\\\n7&\\Pb{2}&&&\\\\\n8&\\Pb{2}&&(2,\\vbn{1})P&\\\\\n9&\\Pb{2}&&(\\vbn{1})\\Pb{0}&\\\\\n10&\\Pb{2}&&(\\vbn{1})\\Pb{0}&\\\\\n11&\\Pb{2}&&&\\\\\n12&\\Pb{2}&&&(2)\\\\\n13&\\Pb{2}&&&\\\\\n\\end{array}$$\nOnce again, the column labelled $u^i$ is the $2i$th column, and shifts\nalong the diagonal have as reference point where this column meets the\nrelevant diagonal. \n\nWe take local cohomology, again remembering that $H^s_{\\Jb_n}$ is\nshifted down by $s$, which changes the diagonal by $s$. For example, on the 7-diagonal, $\\Pb{2}$ comes from the 7-diagonal in\n$BB$, whereas the $\\Pb{0}^{\\vee}(-5\\rho)$ comes from the 2nd local\ncohomology of the entry $(\\vbn{1})\\Pb{0}$ on the $9$-diagonal; the local\ncohomology of $\\Pb{0}$ is $\\Pb{0}^{\\vee}(-4\\rho)$, this is shifted by\na further $-2\\rho$ from the change of diagonal, and $+\\rho$ because of\nthe $\\vbn{1}$. \n$$\\begin{array}{c|cccc}\n&H^*_{(\\vb_1,\\vb_2)}(BB)&(n=2)&&\\\\\n\\hline\nd&1&u&u^2&u^3\\\\\n\\hline\n-2&{\\color{teal}P^*(-6\\rho)}&&&\\\\\n-1&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}&&&\\\\\n0&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}&&&\\\\\n1&&&&\\\\\n2&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&{\\color{teal}P^*(-6\\rho)}&&\\\\\n3&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n4&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n5&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n6&&& {\\color{brown}\\Pb{1}^{\\vee}(-5\\rho)}\\,\\oplus\\, {\\color{teal}P^*(-6\\rho)}&\\\\\n7&\\Pb{2}&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n8&\\Pb{2}&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n9&\\Pb{2}&&&\\\\\n10&\\Pb{2}&&&{\\color{teal}P^*(-6\\rho)}\\\\\n11&\\Pb{2}&&&\\\\\n12&\\Pb{2}&&&\\\\\n13&\\Pb{2}&&&\\\\\n\\end{array}$$\n\nWe have colored again $H^1$-groups in brown and now also $H^2$-groups in teal. We will see below that there are non-trivial extensions on the 2- and 10-diagonals, and that\nthere are differentials in the local cohomology spectral sequence from\nthe 7-, 8- and 9-diagonals (differentials go from the $d$-diagonal to\nthe $(d-1)$-diagonal). \n\n\\subsection{Moving from the basic block $BB$ to the negative block $NB$}\nMoving from $BB$ to $NB$ only affects the $0$ column, where in each\ncase $M$ is replaced by $\\ker (M \\longrightarrow \\mathbb{F}_2)=(2)M$. In effect this\nreplaces $\\Pb{n}$ by $0$. It also adds on a new $(-1)$-column \nof $\\Pb{n}=\\mathbb{F}_2$ going up from the $\\sigma$ row. We resist the temptation\nto display a table for $NB$ explicitly, but note that\n$NB=\\Gamma_{(a)}BB$ as for $k\\R$. \n\n\\subsection{Gorenstein duality}\n\\label{subsec:GorDBPRn}\nWith the above data in mind, we may consider the $d$-diagonal $BB_d$,\nwhere the lowest value of $d$ is 0 and the highest is $N=4(2^n-1)$. If\nwe ignore the difference\nbetween $BB$ and $NB$ (which is at most $\\mathbb{F}_2$ in any degree) we find\napproximately that $BB_d$ has a relationship to $BB_{N-d}$, namely \nsomething like an equality\n$$H^n_{\\Jb_n}(BB_d)^*=BB_{N-d}. $$\nThere are various ways in which this is inaccurate and needs to be modified. \nFirstly, if the local cohomology of $BB_d$ is entirely in cohomological \ndegree $n-\\eps $ with $\\eps \\neq 0$, there will be a shift of $\\eps$\n(if it is in several degrees there is a further\ncomplication). Secondly, Anderson duality introduces a shift of 1 diagonal if\napplied to torsion modules. Thirdly, we have seen that there may be\nextensions between these local cohomology groups, sometimes removing\n$\\Z$-torsion. Finally, there may be differentials. \n\nIn fact all of these effects are `small' in the sense that the growth\nrate along a diagonal is bounded by a polynomial of degree $n-1$. \nEncouraged by this, if we ignore all of these effects we see \nthat $BB$ is a Gorenstein module in the sense that the reverse-graded \nversion is equivalent to \nthe dual of its local cohomology.\n$$H^n_{\\Jb_n}(BB)^*=\\mathrm{rev}(BB). $$ \n\nThis is rather as if there is a cofibre sequence \n$$S\\longrightarrow BP\\R \\langle n \\rangle\\longrightarrow Q$$\nwith $S$ Gorenstein and $Q$ a Poincar\\'e duality algebra of formal dimension\n$N=2(1-\\sigma)(2^n-1)$. \n\n\n\n\n\n\\section{The local cohmology spectral sequence for $tmf_1(3)$}\n\\label{sec:tmfotlcss}\nWe examine the local cohomology spectral sequence and Gorenstein duality\nin more detail for $tmf_1(3)$. Actually, our calculations are equally valid for all forms of $BP\\R\\langle 2\\rangle$, but we prefer the more evocative name $tmf_1(3)$ of the most prominent example. More of the general features are visible for $tmf_1(3)$\nthan for $k\\R$.\n\n As usual we will implicitly localize everywhere at $2$ (although for $tmf_1(3)$ itself it would actually suffice to just invert $3$).\n\n\\subsection{The local cohomology spectral sequence}\nWe make explicit the implications for the coefficient ring, both\ncomputationally and structurally. Writing $\\bigstar$ for\n$RO(\\Ctwo)$-grading as usual, the spectral sequence takes the\nfollowing form. \n\n\\begin{prop} \n\\label{prop:tmfotlcss}\nThere is a spectral sequence of $tmf_1(3)^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{\\Jb_n}(tmf_1(3)^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-8-2\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{tmf_1(3)}).$$\nThe homotopy of the Anderson dual is easily calculated\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(tmf_1(3)^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}\\Z^{tmf_1(3)} \\longrightarrow \\mathrm{Hom}_{\\Z}(tmf_1(3)^{\\Ctwo}_{-\\alpha}, \\Z) \\longrightarrow 0. $$\nThe local cohomology spectral sequence has some non-trivial differentials. \n\\end{prop}\n\n\\subsection{The ring $\\protect tmf_1(3)^{\\Ctwo}_{\\bigstar}$}\\label{sec:tmfgroups}\n\nThe ring $tmf_1(3)^{\\Ctwo}_{\\bigstar}$ is approximately calculated in \\cite{HM} and is more precisely desribed as \n$$BB[U] \\oplus U^{-1}NB[U^{-1}]$$\nas at the end of Section \\ref{sec:BPRnC2} with $n=2$. We already tabulated $BB$ in Section \\ref{subsec:tmfotloccoh}, but we want also want to display a bigger chart of $\\pi_{\\bigstar}^{C_2}tmf_1(3)$ as Figure \\ref{fig:tmf13} to give the reader a feeling of how the blocks piece together. \n\nA black diagonal line means a copy of $P$ when it starts in a box, a\ncopy of $(2)P$ when it starts in a small circle, a copy of\n$(2,\\vb_1)P$ when it starts in a dot and a copy of $(2,\\vb_1,\\vb_2)$\nwhen it starts in a big circle. A red diagonal line means a copy of\n$\\overline{P}_0$ and a green diagonal line a copy of $\\overline{P}_1$. A red dot is a copy of $\\F_2 = \\overline{P}_2$. \n\n\n\\begin{center}\n\\begin{figure}\\includegraphics{tmf13}\n\\caption{The homotopy of $tmf_1(3)$ \\label{fig:tmf13}}\n\\end{figure}\n\\end{center}\n\n\\subsection{Local cohomology}\nWe are calculating local cohomology with respect to the \nideal $\\Jb_2=(\\vbn{1}, \\vbn{2})$ so that we only need to consider $tmf_1(3)^{\\Ctwo}_{\\bigstar}$ as a\n$\\Z [\\vbn{1}, \\vbn{2}]$-module. As such it is a sum of suspensions of the blocks\n$BB$ and $NB$, so we just need to calculate the local cohomology of\nthese. This was described in Section \\ref{sec:BPRnlcss} above. Here we will\nsimply describe the extensions and the behaviour of the local\ncohomology spectral sequence. \n\nThe basis of this discussion are the tables of $BB$ and $GBB$ from Subsection\n\\ref{subsec:tmfotloccoh} together with the analogues for $NB$ and\n$GNB$. Although these are organized by diagonal, Figure \\ref{fig:GBBtmf13}\ndisplaying $BB, GBB, U^{-1} NB$ and $U^{-1}GNB$ may help visualize the \nway the modules are distributed along each diagonal. The vertical lines denote\nmultiplication by $a$ and the dashed vertical line is an exotic\nmultiplication by $a$ that is not visible on the level of local\ncohomology. The green diamond does not denote a class, but marks the\npoint one has to reflect (non-torsion classes) at to see Anderson\nduality. Torsion classes are shifted after reflection by $-1$ (i.e.,\none step horizontally to the left). \n\n\\begin{center}\n\\begin{figure}\\includegraphics{GBBtmf13}\n\\caption{Gorenstein duality for $tmf_1(3)$ \\label{fig:GBBtmf13}}\n\\end{figure}\n\\end{center}\n\n\n\nThe strategy is to take the known subquotients from the local\ncohomology calculation, and resolve the extension problems using Gorenstein\nduality. \n\n\n\n\n\\begin{prop}\nWe have an isomorphism \n$$\\pi_{\\bigstar}^{C_2}\\Gamma_{\\Jb_2}tmf_1(3) \\cong GBB[U]\\oplus U^{-1}GNB[U^{-1}],$$\nwhere $GBB$ and $GNB$ are described in the following. We will simultaneously describe what differentials and extensions in the local cohomology spectral sequence caused the passage from $H^*_{\\Jb_2}(BB)$ and $H^*_{\\Jb_2}(NB)$ to $GBB$ and $GNB$ respectively. \n\n(i) The $\\Z [\\vbn{1}, \\vbn{2}]$-modules along the diagonals in $GBB$ are as\nfollows. \n$$\\begin{array}{c|cl}\n&GBB&(n=2)\\\\\n\\hline\ni&\\mbox{Module}&\\mbox{Top degree}\\\\\n\\hline\n-2&P^*&-6-4\\sigma\\\\\n-1&\\Pb{0}^{\\vee}&-6-5\\sigma\\\\\n0&\\Pb{0}^{\\vee}&-6-6\\sigma\\\\\n1&0&\\\\\n2&[(2,\\vbn{1})P]^*&-4-6\\sigma\\\\\n3&\\Pb{1}^{\\vee}&-4-7\\sigma\\\\\n4&\\Pb{1}^{\\vee}&-4-8\\sigma\\\\\n5&\\Pb{1}^{\\vee}&-4-9\\sigma\\\\\n6&[(2,\\vbn{1})P]^*&-2-8\\sigma\\\\\n7&(\\vbn{1}, \\vbn{2})\\Pb{0}&-2-9\\sigma\\\\\n8&(\\vbn{1}, \\vbn{2})\\Pb{0}&-2-10\\sigma\\\\\n9&0&\\\\\n10&[(2, \\vbn{1}, \\vbn{2})P]^*&0-10\\sigma\\\\\n10+k\\geq 11&\\mathbb{F}_2&0-(10+k)\\sigma\\\\\n\\end{array}$$\nThere are three non-trivial differentials\n$$d_2: H^0_{\\Jb_2}(BB)\\longrightarrow H^2_{\\Jb_2}(BB)$$\nfrom the groups at $-7\\sigma, -8\\sigma, -9\\sigma$ to the groups at \n$-7\\sigma-1, -8\\sigma -1, -9\\sigma-1$, which have affected the values\non the 6-, 7-, 8- and 9-diagonals in the table. \n\nThe extensions \n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1})P]^* \\longrightarrow \\mathbb{F}_2 [\\vb_2]^{\\vee}\\longrightarrow 0$$\non the 2-diagonal and the 6-diagonal are Anderson dual to the defining short exact sequence\n$$0\\longrightarrow (2,\\vbn{1})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 [\\vb_2]\\longrightarrow 0$$\nin the following sense: The Anderson dual of the latter exact sequence is a triangle\n$$\\mathbb{F}_2[\\vb_2]^* \\to P^*\\to [(2,\\vbn{1})P]^* \\to \\Sigma\\mathbb{F}_2[\\vb_2]^* \\cong \\mathbb{F}_2[\\vb_2]^{\\vee},$$\nwhich induces (on homology) the extensions above. \nThe extension \n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1}, \\vbn{2} )P]^* \\longrightarrow \\mathbb{F}_2 \\longrightarrow 0$$\non the 10-diagonal is Anderson dual to the short exact sequence\n$$0\\longrightarrow (2,\\vbn{1}, \\vbn{2})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 \\longrightarrow 0.$$\n\n(ii) The $\\Z [\\vbn{1}, \\vbn{2}]$-modules along the diagonals in $GNB$ are as\nfollows (take the direct sum of the two entries for the $(-2)$-,\n$(-1)$-, $0$- $1$- and $2$-diagonals) \n$$\\begin{array}{c|cl}\n&GNB&(n=2)\\\\\n\\hline\ni&\\mbox{Module}&\\mbox{Top degree}\\\\\n\\hline\n-k\\leq -3&\\mathbb{F}_2&-1-k\\sigma\\\\\n-2&P^*, \\mathbb{F}_2 &-6-4\\sigma, -1+\\sigma\\\\\n-1&\\Pb{0}^{\\vee}, \\mathbb{F}_2&-6-5\\sigma, -1+0\\sigma\\\\\n0&\\Pb{0}^{\\vee}, \\mathbb{F}_2 &-6-6\\sigma, -1-\\sigma\\\\\n1&\\mathbb{F}_2 &-1-2\\sigma\\\\\n2&P^*, \\Pb{1}^{\\vee}&-4-6\\sigma, -1-3\\sigma\\\\\n3&\\Pb{1}^{\\vee}&-1-4\\sigma\\\\\n4&\\Pb{1}^{\\vee}&-1-5\\sigma\\\\\n5&\\Pb{1}^{\\vee}&-1-6\\sigma\\\\\n6&[(2,\\vbn{1})P]^*&-1-7\\sigma\\\\\n7&\\Pb{0}^{\\vee}&-1-8\\sigma\\\\\n8&\\Pb{0}^{\\vee}&-1-9\\sigma\\\\\n9&0&\\\\\n10&P^*&0-10\\sigma\\\\\n\\end{array}$$\n\nThe extension\n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1})P]^* \\longrightarrow \\mathbb{F}_2 [v_2]^{\\vee}\\longrightarrow 0$$\non the 6-diagonal is Anderson dual to the short exact sequence\n$$0\\longrightarrow (2,\\vbn{1})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 [v_2]\\longrightarrow 0.$$\n\\end{prop}\n\n\n\n\\begin{proof}\nWe first note that the contributions from the different blocks do not\ninteract. Indeed, the only time that different blocks give\ncontributions in the same degree comes from the $\\mathbb{F}_2 [a]$ towers of\n$BB$, and one class in that degree is divisible by $\\vbn{1}$ or $\\vbn{2}$ and not killed\nby both $\\vbn{1}$ and $\\vbn{2}$. We may therefore \nconsider the blocks entirely separately.\n \nThe block $GBB$ comes from the local cohomology of $BB$ in the sense\nthat there is a spectral sequence\n$$H^*_{\\Jb_2}(BB)\\Rightarrow GBB .$$\nThus there is a filtration \n$$GBB=GBB^0\\supseteq GBB^1\\supseteq GBB^2\\supseteq GBB^3=0$$\nwith \n$$0\\longrightarrow GBB^0\/GBB^1 \\longrightarrow H^0_{\\Jb_2}(BB) \\stackrel{d_2}\\longrightarrow\n\\Sigma^{-1}H^2_{\\Jb_2}(BB) \\longrightarrow \\Sigma^1 GBB^2\\longrightarrow 0$$\nand \n$$GBB^1\/GBB^2\\cong \\Sigma^{-1} H^1_{\\Jb_2}(BB). $$ \n\nThe block $GNB$ comes from the local cohomology of $NB$ in a precisely\nanalogous way.\n\nMost questions about module structure over $BB[U]$ are resolved by\ndegree. The remaining issues are resolved by\nusing Gorenstein duality. \n\n\nReferring to the table for $H^*_{\\Jb_2}(BB)$ in Subsection\n\\ref{subsec:tmfotloccoh}, the first potential extension is on the\n2-diagonal. Using Gorenstein duality to compare with $NB_{\\delta=8}$ \nwe see that the actual extension on the 2-diagonal of $GBB$ is \n$$0\\longrightarrow P^*\\longrightarrow [(2,\\vb_1)P]^*\\longrightarrow \\Pb{1}^{\\vee} \\longrightarrow\n0, $$\nwhere we have shifted the modules so they all have top degree 0. \nThere is an additive extension on the 10-diagonal by\nreference to the Anderson dual. \nFinally the three non-zero $d_2$ differentials from $-1-k\\sigma$ for\n$k=7,8$ and $9$ are necessary for connectivity (this removes the \nneed to discuss the possible extensions on the 7- and 8-diagonals). \n\n\nThe situation is rather similar for $GNB$. We will not explicitly\ndisplay $NB$ since the only effect (apart from the addition of \n$\\mathbb{F}_2 [a]^{\\vee}$) is on the first column, where a\nmodule is replaced by the kernel of a surjection to $\\mathbb{F}_2$. It is\nperhaps worth displaying $H^2_{\\Jb_2}(NB)$, where we leave out the big $\\mathbb{F}_2[a]^{\\vee}$-tower in $H^0_{\\Jb_2}NB$. We will color again $H^1$-groups in brown and $H^2$-groups in teal.\n$$\\begin{array}{c|cccc}\n&H^*_{\\Jb_2}(NB)&(n=2)&&\\\\\n\\hline\ni&1&u^2&u^4&u^6\\\\\n\\hline\n-2&{\\color{teal}P^*(-6\\rho)}&&&\\\\\n-1&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}\\oplus {\\color{brown}\\Pb{2}}&&&\\\\\n0&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}\\oplus {\\color{brown}\\Pb{2}}&&&\\\\\n1&{\\color{brown}\\Pb{2}}&&&\\\\\n2&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&{\\color{teal}P^*(-6\\rho)}&&\\\\\n3&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n4&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n5&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n6&&& {\\color{brown}\\Pb{1}^{\\vee}(-5\\rho)}\\oplus {\\color{teal}P^*(-6\\rho)}&\\\\\n7&&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n8&&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n9&&&&\\\\\n10&&&&{\\color{teal}P^*(-6\\rho)}\\\\\n11&&&&\\\\\n12&&&&\\\\\n13&&&&\\\\\n\\end{array}$$\n In this case all extensions are split, except for the one on the\n6-diagonal and there are no differentials. The $a$ multiplications\nin the $\\mathbb{F}_2 [a]^{\\vee}$ tower are clear from Gorenstein duality and\nthe $a$-tower $\\mathbb{F}_2 [a]$ in $BB$.\n\\end{proof}\n\n\\begin{remark}\n(i) Summarizing the way a diagonal $BB_{\\delta}$ contributes to $NB$ as in \n$$H^*_{\\Jb_2}(BB_{\\delta})^*\\sim NB_{\\delta'}$$ \nas sketched in Subsection \\ref{subsec:GorDBPRn}. We have \n\n$$\\begin{array}{|cc||cc|}\n\\delta &\\delta' s.t. H^*_{\\Jb_2}(BB_{\\delta})^*\\sim\nNB_{\\delta'}&\\delta &\\delta' s.t. H^*_{\\Jb_2}(NB_{\\delta})^*\\sim BB_{\\delta'}\\\\\n\\hline\n0&12&0&12\\\\\n1&10&1&10\\\\\n2&9&2&9\\\\\n3&8&3&8\\\\\n4&8,6&4&8,6\\\\\n5&5&5&5\\\\\n6&4&6&4\\\\\n7&2&7&.\\\\\n8&4,3&8&4\\\\\n9&2&9&2\\\\\n10&1,0&10&1\\\\\n11&0&11&.\\\\\n12&0&12&0\\\\\n\\hline\n\\end{array}$$\n\nBecause most of the modules are $2$-torsion the most common pairing is\nbetween $\\delta$ and $11-\\delta$ rather than between \n$\\delta$ and $12-\\delta$ as happens for the main $U$-power diagonals. \n\n(ii) We also note as before that since $NB=\\Gamma_{(a)}BB$, we have\n$$\\Sigma^{6+4\\sigma}\\Gamma_{(\\vb_1, \\vb_2)}BB\\sim (\\Gamma_{(a)} BB)^*$$\n(where we have written $\\sim$ rather than $\\simeq$ in recognition of\nthe differentials) and \n$$\\Sigma^{6+4\\sigma}\\Gamma_{(\\vb_1, \\vb_2, a)} BB\\simeq BB^*, $$\nwith the second stating that $BB$ is Gorenstein of shift $-6-4\\sigma$\nfor the ideal $(\\vb_1, \\vb_2, a )$. \n\\end{remark}\n\n\n\\vspace{1cm}\n\\addtocontents{toc}{\\vspace{\\normalbaselineskip}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Intro}\nEfficient telescope scheduling is essential to maximize the scientific output of survey telescopes. \nOptimizing a survey's scientific merit function requires scheduling decisions that consider slew rate, sky brightness, source location, transparency and extinction, and many other factors. For surveys such as LSST that will scan the entire accessible sky \\citep{Ivezic2014}, a determination of the sky brightness as a function of azimuth, elevation and passband is an important factor in crafting an optimal sequence of observations. Contributions to the night sky include unresolved or diffuse celestial sources, emission from OH molecules in the upper atmosphere, zodiacal light, man-made light pollution, and moonlight that scatters from clouds and from the constituents of the atmosphere. Some of these contributions to the night sky are stable over time and do not impact the order in which we observe fields. Other \ncontributions to the night sky are time-variable but deterministic. The monthly \nvariation due to moonlight falls in this category, for cloud-free conditions. Other \ncontributions to sky brightness, such as variable OH emission \\citep{High2010} and moonlight scattering from clouds, are more stochastic in nature. \n\nUnderstanding the brightness of the cloudless moonlit sky in the LSST bands is one key component in scheduling decisions. To make predictions of potential future performance, LSST has developed an operations simulator to study different scheduling algorithms \\citep{OpSim}. \nThus far, the LSST operations simulator has used historical weather records and measures of the atmospheric conditions for the Cerro Pachon site (taken over a 10-year time period). \nThe operations simulator generates a sky model that predicts sky brightness based on the Krisciunas and Schaefer model \\citep{KrSc1991}, which is further discussed below. It also simulates atmospheric seeing and cloud coverage as a function of time. This information is used to estimate the efficiency of the survey for different candidate scheduling algorithms. As LSST advances into the construction phase, and eventually into full operation, we need a higher fidelity determination of the brightness of the cloudless \nnight sky. This will be augmented with all-sky-camera data \\citep{AllSkyCamera} to make real-time, condition-dependent adjustments to the sequencing of LSST observations. \n\nWe define the lunar contribution to sky brightness as the difference between the observed sky brightness (in units of magnitudes per square arc sec) with the Moon above the horizon and the moonless sky brightness, for a given the phase of the lunar cycle. From the standpoint of scheduling decisions, what matters most is the {\\it relative} lunar brightness variation across the accessible sky, and so our primary goal in this paper is to determine this spatial structure, using the sun as a proxy for the moon. This allows us to obtain high signal-to-noise data, without complications from other contributions to sky brightness. \n\nAn {\\it ab initio} computation of the lunar sky illumination is complicated due to multiple scattering effects. Sunlight reflects off the moon, and a portion of this light is scattered towards the Earth. This light impinging on the top of the atmosphere is then scattered by molecules and aerosols, and some is absorbed. The moonlight can be scattered multiple times, including off of the ground, before it reaches the telescope pupil. An empirical measurement is arguably more secure than a radiative transfer calculation that must make assumptions about the size, shape, and vertical distribution of aerosols. \n\nWalker developed \\citep{Walker1987} a scattered moonlight model that included a table of sky brightness in five photometric bands, at five different moon phases. It did not account for the positions of the Moon or observation target, and was measured during solar minimum. Because of these shortcomings, it is not accurate enough for current and future telescope operations. Later, Krisciunas and Schaefer used an empirical fit to 33 observations taken in the V-band taken at the 2800\\,m level of Mauna Kea, resulting in an accuracy between 8\\% and 23\\% if not near full Moon \\citep{KrSc1991}. This model predicted the moonlight as a function of the Moon's phase, the zenith distance of the Moon, the zenith distance of the sky position, the angular separation of the Moon and sky position, and the band's atmospheric extinction coefficient. More recently, a spectroscopic extension of this model was used to fit sky brightness data from Cerro Paranal \\citep{Noll2012}. This treatment includes all relevant components, such as scattered moonlight and starlight, zodiacal light, airglow line emission and continuum, scattering and absorption within the Earth's atmosphere, and thermal emission from the atmosphere and telescope. This model was recently updated with an observed solar spectrum, a lunar albedo fit, and scattering and absorption calculations \\citep{Jones2013}. Winkler et al. characterized the nighttime sky brightness profile under a variety of atmospheric conditions using measurements from the South African Astronomical Observatory soon after the Mount Pinatubo volcanic eruption in 1991 \\citep{WiWy2013}.\nOur goals in this paper are more limited, as we are primarily interested in the \nspatial structure and spectrum of the scattered moonlight component of the night sky. \n\nBased on this discussion, it is clear that models for scattered moonlight are very complicated. This motivates our attempt to empirically determine the relative sky brightness as a function of lunar phase, and its dependence on the positions of the target and the moon. We measured the solar sky brightness as a function of angle between sky location and the sun, as well as zenith angles of the sun and the telescope. We argue that this is useful because up to wavelength-dependent lunar albedo factors, the sky illumination pattern that the moon casts has the same functional form as from the sun in the daytime. \n\nWe have made measurements of the daytime sky brightness at Cerro Pachon, the LSST site in Chile, with an array of six photodiodes with filters in \nthe {\\it u, g, r, i, z,} and {\\it y} bands. \nThere is an extensive history of daytime sky brightness measurements. In particular, the \nangular and wavelength dependence of the observed solar scattering can be used to deduce properties of atmospheric aerosols and precipitable water vapor. \nWe use a similar measurement scheme to the AERONET remote sensing aerosol monitoring network \\citep{Holben1998}. The AERONET sky brightness data are taken in optical passbands that differ from those that LSST will use. Rather than invoke a set of \ncolor transformations to convert from AERONET into LSST bands, here we make a \ndirect measurement of sky brightness in the LSST passbands. \n\nWe describe the apparatus in section \\ref{sec:apparatus}. Measurements and analysis are presented in section \\ref{sec:results}. We conclude with a discussion of topics for further study in section \\ref{sec:Conclusion}. \n\n\\section{Apparatus}\n\\label{sec:apparatus}\n\n\\begin{figure}[t]\n \\includegraphics[width=6.0in]{plots\/photodiode_drawing.pdf}\n \\caption{Sketch of the photodiode portion of the apparatus. The photodiode mounts include a photodiode, an iris, a filter, and a baffle tube.\n \\label{fig:apparatus}\n\\end{figure}\n\nFig.~\\ref{fig:apparatus} shows a sketch of the photodiode mount, which is identical \nfor all six channels except for the interference filters that define the passbands. \nLight enters a 50\\,mm inner diameter cylinder, 152.4 mm long, that serves to block off-axis stray light. The 50\\,mm diameter filters are placed at the base of this baffle tube. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=6.0in]{plots\/filter.pdf}\n\\caption{Photon sensitivity function curves for the photodiodes (upper) used in this experiment, and \nthe expected photon sensitivity function for LSST (lower curves, due to more complex optical system). From left to right the bands are $u,g,r,i,z$ and $y$. We use this information\nto make a throughput correction when predicting scattered moonlight backgrounds for LSST. The Astrodon filters we used for the photodiodes are designed to avoid the water band at 940 nm. }\n\\label{fig:QE}\n\\end{center}\n\\end{figure}\n\nWe used ``Generation 2 Sloan Digital Sky Survey (SDSS)'' {\\it u,g,r,i,z,y} filters from Astrodon \\citep{Astrodon}. \nFigure~\\ref{fig:QE} shows their transmission spectra as well as the current-design LSST filters \\citep{Ivezic2014}, for comparison. The Astrodon filters we used are essentially flat-topped, with minimal leakage or in-band ripple. An adjustable iris (Thor Labs SMD12C) sits behind the interference filter. We found we could operate with these irises set to their maximum opening diameter of 12\\,mm, for all six passbands. \nThe only other transmissive optical element that lies between the Si and the sky is a quartz window in front of the photodiode. \n\nThe photodiodes are SM1PD2A cathode-grounded Si UV-enhanced photodiodes, obtained from Thor Labs. The photodiodes have a 10\\,mm x 10\\,mm active area behind a 9\\,mm diameter input aperture. The only other transmissive optical element that lies between the Si and the sky is a quartz window in front of the photodiode. The etendue of the system is established by the combination of the 12\\,mm diameter iris and the 9\\,mm circular photodiode input aperture. These two circular apertures are separated by a distance of 60\\,mm. \n\n\\subsection{Etendue of the photodiode plus tube system, in comparison to an LSST pixel}\n\nAs seen from the plane of the diode aperture, the full-angle subtended by the adjustable iris is then $2\\,\\textrm{arctan}(\\frac{6}{60})=11.4^\\circ$, which subtends a solid angle of \n$\\Omega_\\textrm{diode}=2\\,\\pi\\,(1-\\cos(\\frac{11.4}{2}))=3.11\\times10^{-2}$ steradians. For comparison, \na pixel on LSST subtends 0.2 arcsec on a side, for a solid angle of $7.4\\times10^{-13}$\\,steradians\/pixel.\n\nIf we consider the iris as establishing the field of view of the photodiode system, then the \naperture in front of the diode determines the sensor's unvignetted collecting area, where $A_\\textrm{photodiode}=\\pi\\,(4.5 \\times 10^{-3} \\textrm{m})^2 = 6.36\\times10^{-5}\\,\\textrm{m}^2$. This amounts to computing the overlap of the sensor and iris apertures. For comparison, the effective collection area of LSST is equivalent to a diameter of 6.5\\,m, for a collection area of $A_\\textrm{LSST}=\\pi\\,(\\frac{6.5\\textrm{m}}{2})^2=33.2\\,\\textrm{m}^2$. The ratio of the etendue of an LSST pixel to the photodiode is then $R=\\frac{A_\\textrm{LSST} \\Omega_\\textrm{LSST pixel}}{A_\\textrm{photodiode} \\Omega_\\textrm{photodiode}}=1.23\\times 10^{-5}$.\n\n\n\nThe interpretation of the data will benefit from knowing the ratio between the instrumental response function of the photodiode system and LSST. Table~\\ref{tab:throughputs} compares the band-integrated system throughput figure for the photodiode system (the Thor labs QE times the Astrodon filter response) with two versions of the LSST throughput. LSST is considering using CCDs from two vendors, e2v and ITL, and these have somewhat different quantum efficiency curves. We have, therefore, provided in Table~\\ref{tab:throughputs} the results from integrating over the response functions (including in the LSST case the three reflections, the obscuration, the filter and corrector transmissions, and the detector QE) at a spacing of one nm. The units in Table~\\ref{tab:throughputs} are nm, and can be interpreted as the sensitivity-weighted equivalent width of the respective filters. Taking the ratio of these numbers, passband by passband, allows us to scale the photodiode measurements to anticipated values on the LSST focal plane. \n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nband & diodes & LSST with ITL & LSST with e2v & $\\left [ \\frac{T_{ITL}} {T_{diodes}} \\right ]$ & $\\left [ \\frac{T_{e2v}} {T_{diodes}} \\right ]$ \\\\\n\\hline\nu & 33.8 & 20.6 & 15.3 & 0.61 & 0.45 \\\\\ng & 99.0 & 61.3 & 65.4 & 0.62 & 0.66 \\\\\nr & 93.2 & 60.3 & 62.9 &0.65 & 0.67 \\\\\ni & 106.7 & 53.7 & 53.2 & 0.50 & 0.50 \\\\\nz & 155.2 & - & - & - & - \\\\\nzs & 65.8 & 44.3 & 43.3 & 0.67 & 0.66 \\\\\ny & 69.5 & 27.9 & 27.2 & 0.40 & 0.40 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{System throughput values. The first three columns are the integral of the system response function at 1 nm spacings, for each passband in the different systems. \nThe last two columns show ratios of the LSST\nthroughput to that of the diodes. The diode instrument has no reflective optics \nand a minimum of air-glass interfaces, whereas LSST has three reflections from \naluminum as well as a three element corrector. Also, the photodiodes are considerably thicker than the LSST CCDs, and have enhanced UV sensitivity. This accounts for the \nincreased diode throughput compared to the LSST system.}\n\\label{tab:throughputs}\n\\end{table}%\n\n \n\nThe photodiodes are connected via coaxial cable to a set of manual switches that feed a selected one of the six signals to a Thor Labs model PDA200C photocurrent amplifier, which produces a $\\pm$ 10\\,V signal proportional to the current from the selected photodiode. This signal was connected to an Arduino Uno, which digitizes this signal with a 10 bit A\/D converter. The Arduino was connected to a serial port on the data collection computer.\n\nThe six photodiode tubes are mounted on a Celestron model CG-5 equatorial telescope mount, which is controlled by external connection to a laptop computer. Stellarium (\\cite{Stellarium}) is used to control the telescope mount pointing. The mount's RA and DEC motors have a precision of 0.05$^\\circ$. The computer registered the right ascension ($\\alpha$) and declination ($\\delta$) for each brightness measurement, along with the photocurrent from each of the six photodiodes. \nThese $\\alpha$ and $\\delta$ measurements were converted to alt-az coordinates for data analysis, since as shown below this is the most natural angular coordinate system for this problem. \n\n\n\\subsection{Sky Scanning Strategy and Angular Coordinates}\n\nAssuming that the scattering properties of the atmosphere are axisymmetric about local vertical, the normalized sky brightness (scaled to the brightness of the illuminating source) depends on three angles: the zenith angle $z_\\textrm{source}$ of the source (sun or moon), the zenith angle $z_\\textrm{tel}$ of the telescope boresight, and the azimuthal angle $\\Delta \\phi$ between the source and the boresight.\n \nAn ``almucantar'' is the line on the sky at the elevation angle of the Sun, at some\ngiven time. An advantage to making sky brightness measurements along an almucantar \nis that the boresight and source elevation angles are constant, and equal. Only their\nazimuthal separation is varied. \nDuring an almucantar measurement, observations are made at the solar elevation angle through 360$^\\circ$ of azimuth. The almucantar sweep is a special case of a constant-zenith-angle scan, which is our favored data collection method. The range of scattering angles along an almucantar decreases as the solar zenith angle decreases; thus almucantar sequences made at airmass of 2 or more achieve maximum scattering angles of 120$^\\circ$ or larger.\n\nWe elected to obtain our sky brightness data in a succession of constant-zenith angle scans, taking a data point every 45$^\\circ$ of azimuth, except near the zenith. This gives us 8 data points in azimuth at each telescope zenith angle. We obtained data at zenith angles of 0, 30, 45, 60, and 75$^\\circ$, and along the almucantar, over the course of the day, in each passband. The resulting daytime sky brightness (DSB) data by passband, $DSB(z_\\textrm{source}, z_\\textrm{tel}, \\phi, \\textrm{filter})$, comprise our measurement. We generate an all-sky map of sky brightness. The data are processed as follows. For each solar zenith angle, we generate an all-sky map of brightness vs. position. We then repeat for different values of solar zenith angle. We make a polynomial fit to brightness vs. altitude and delta-azimuth. This is a map of relative night sky brightness if the moon were at the location of the sun, up to an overall scale factor per passband. We can use the geometry of the photodiode tube to compute number of photons per square arcsec per square meter and then scale the value by about 14 magnitudes to get the lunar contribution. The scaling factors are computed in the next section. We can then compute an estimate for other lunar phases, based on the lunar phase function. Of course, the actual sky brightness is a combination of the lunar contribution, which we compute, plus other factors, which depend on the instrument, plate scale, integration time, etc. as well as solar cycle and site characteristics. \n \n\\subsection{Scaling from Solar to Lunar Illumination}\n\nKeiffer and Stone (\\cite{KiSt2005}, hereafter K\\&S)) describe how to scale between solar \nillumination and lunar\nillumination at the top of the atmosphere, depending on both reflection geometry\nand wavelength. The ratio $R$ of the lunar to solar irradiance at the top of the atmosphere \nis given by\n$R(\\lambda,g)=A(\\lambda, g) \\left [ \\Omega_M\/\\pi \\right ] $, where $\\Omega_M$ \nis the solid angle subtended by the moon, and $A(\\lambda,g)$ is a wavelength \ndependent scattering function that depends on the angle, g, between the vectors \nfrom the moon to the Earth and the moon to the sun. We\nhave assumed nominal values for the sun-moon and moon-earth distances. \nThe geometrical dilution factor is 6.42\/$\\pi \\times 10^{-5} = 2.04 \\times 10^{-5}$, or 11.72 magnitudes. \n\nTo correct for the wavelength-dependent and phase-angle-dependent lunar scattering function, we took the parametric description of $A(\\lambda,g)$ provided in K\\&S, integrated across our passbands (note: truncated u band at 350, no data bluer), to determine the scattering-dependent magnitude differences between sunlight and moonlight at the top of the atmosphere, for different lunar phases. In order to avoid the sharp peak in reflection due\nto the ``opposition effect'', we limited the range of phase angles to $|g| > $ 2 degrees.\n\nThe additional attenuation from lunar scattering as a function of passband at \nfull moon (taken here to be our minimum scattering\nangle of 2 degrees) is listed in Table \\ref{tab:fullmoon}. We obtained these values by numerically computing \n\n\\begin{equation}\n\\Delta \\textrm{mag}_i (g) = -2.5 \\textrm{log}_{10} \\left( \\frac{\\int A(\\lambda,g) T_i(\\lambda) {\\rm d \\lambda}} {\\int T(\\lambda) {\\rm d \\lambda}} \\right)\n\\label{eq:fullmoon}\n\\end{equation}\n\n\\noindent\nwhere $T_i (\\lambda)$ is a top-hat approximation of filter $i$. \n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nBand & $\\Delta \\textrm{mag}(g=2^\\circ)$ & $\\Delta \\textrm{mag}(g=10^\\circ)$ & $\\Delta \\textrm{mag}(g=45^\\circ)$ & $\\Delta \\textrm{mag}(g=90^\\circ)$ \\\\\n\\hline\n$u$ & 2.60 & 3.05 & 4.06 & 5.52\\\\\n$g$ & 2.36 & 2.78 & 3.77 & 5.19 \\\\\n$r$ & 2.10 & 2.50 & 3.45 & 4.84 \\\\\n$i$ & 1.92 & 2.31 & 3.23 & 4.59 \\\\\n$z$ & 2.17 & 2.57 & 3.53 & 4.92 \\\\\n$y$ & 1.72 & 2.12 & 2.91 & 4.31 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Irradiance attenuation due to lunar scattering, in the LSST bands, at various lunar phase angles. The illumination from the moon is slightly redder than sunlight, in general. \nThis reddening effect increases as the phase angle increases.}\n\\label{tab:fullmoon}\n\\end{table*}%\n\nBased on these values, since the sky brightness scales linearly with the irradiance provided at the top of the atmosphere, we expect the $r$ band full-moon lunar sky brightness to be a factor of 11.72+2.10=13.82 magnitudes fainter than what we observe in the daytime, if the moon were placed in the same alt-az position as the sun. \n\n\nThis allows us to generate, up to a single passband-dependent overall scale factor that depends on the \neffective etendue of the photodiode tube, the equivalent full-moon sky brightness map\nfor the case where the moon is in the same location as the sun. \nWe simply take the solar-illuminated sky brightness data, and scale all values to the\nr band brightness at the zenith. Then we make a passband-dependent adjustment based\non the color of the reflected sunlight as shown in Table \\ref{tab:fullmoon}.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nFilter & Dark Current [nA] & $1 \\sigma$\\,Uncertainty [nA] \\\\\\hline\\hline\nu & 4.3 & 0.2 \\\\\\hline\ng & 2.1 & 0.3 \\\\\\hline\nr & 5.3 & 0.2 \\\\\\hline\ni & 4.2 & 0.2 \\\\\\hline\nz & 4.6 & 0.2 \\\\\\hline\ny & 2.0 & 0.3 \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Dark current measurement for all six filters. Measurements were taken periodically during data taking to check if there was a significant temperature dependence. Dark current values were found to be approximately the same over the run, and these values are about 100 times smaller than the currents in the sky brightness analysis.\n}\n\\label{tab:DarkCurrent}\n\\end{table}\n\nWe begin by measuring dark current values for each photodiode channel, and the results are displayed in Table~\\ref{tab:DarkCurrent}. The photodiodes have dark current values ranging from 2.0 to 5.3\\,nA with statistical uncertainties between 0.2 to 0.3\\,nA. These dark currents are well under 1\\% of the signal levels from the sky. Because it is a negligible contribution, we ignore the dark current contribution in the analysis that follows.\n\n\\subsection{Observations}\n\nWe obtained sky brightness data from the roof of the ALO building on Cerro\nPachon, (located at S 30:15:06, W 70:44:18) during the daytime on \n2014 Sept 4, 5, 6 and 7. The conditions on Sept 5 were less favorable, \nwith high cirrus clouds in the sky. \nWe cycled through the sky sampling strategy described above, \ntaking 2000 data points in each passband, running through the 6 bands\nin succession. Each data collection period at a fixed pointing lasted about \ntwo minutes, and a full cycle across the sky lasted about an hour. In all, we collected 10 sequences, spanning a range of solar elevations from 20$^\\circ$ to 55$^\\circ$. \n\n\\subsection{Spatial structure of scattered light}\n\nNight sky structure was investigated by \\cite{ChHa1996}, in the context of flat-fielding. The authors are unaware of a comprehensive program to map (and visualize) the sky brightness under variable lunar illumination conditions. In the following, we show the sky brightness dependence on the zenith angle $z_\\textrm{source}$ of the source (sun or moon), the zenith angle $z_\\textrm{tel}$ of the telescope boresight, and the azimuthal angle $\\phi$ between the source and the boresight. We perform fits of sky brightness to the three independent variables and compute the effect on 5 sigma point source detection magnitude for a survey such as LSST. The data allows us to study the color across the sky. We also note how to scale overall brightness in each band as a function of lunar phase.\n\n\\subsection{Spatial Structure of Lunar Sky Brightness}\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nBand & a ($\\times 10^{11}$) & b ($\\times 10^{11}$) & c ($\\times 10^{11}$) & d ($\\times 10^{11}$) & Median Residual ($\\times 10^{11}$) \\\\\n\\hline\nu & 88.5 (6.2) & -0.5 (0.1) & -0.5 (0.1) & 0.4 (0.1) & 5\\\\\ng & 386.5 (34.0) & -2.2 (0.2) & -2.4 (0.2) & 0.8 (0.5) & 13\\\\\nr & 189.0 (32.7) & -1.4 (0.2) & -1.1 (0.2) & 0.8 (0.5) & 11\\\\\ni & 164.8 (33.1) & -1.5 (0.2) & -0.7 (0.2) & 0.6 (0.5) & 12\\\\\nz & 231.2 (62.3) & -2.8 (0.3) & -0.7 (0.4) & 1.4 (0.9) & 21\\\\\nzs & 131.1 (45.6) & -1.4 (0.2) & -0.5 (0.3) & 0.2 (0.6) & 10\\\\\ny & 92.0 (32.7) & -1.3 (0.2) & -0.2 (0.2) & 0.9 (0.5) & 20\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Daytime sky brightness values as function of angle between the point on sky and sun, altitude of the point on the sky, and altitude of sun fit to a plane of the form $a + b x + c y + d y$ in electrons\/s. The zs band values, which are given to approximate the LSST z filter, are computed using Astrodon z minus y. Here, x corresponds to the angle between the point on the sky and the sun, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the sun. The median of the absolute value of the residuals is given by the final column. }\n\\label{tab:results}\n\\end{table*}\n\nThe first result we present is sky brightness as a function of angle between the point on sky and the sun, the altitude of the point on the sky, and the altitude of the sun. The measurements describe a three-dimensional surface corresponding to these parameters. We convert the photodiode output from $\\mu$A to electrons\/s. We fit resulting data to a plane of the form $a + b x + c y + d y$ for each of the six bands. Here, x corresponds to the angle between the point on sky and the sun, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the sun. The coefficients and the standard errors, as well as the median of the residuals for the fits are shown in Table~\\ref{tab:results}. We find the residuals of the fits to be small; these are generally an order of magnitude smaller than the overall scale factor (a). There are a number of notable features. The first is that as $\\phi$ increases, the sky brightness decreases. This in and of itself is not surprising, but the rates of decrease are significant. For example, in the g band, for every 10 degrees that $\\phi$ decreases, the number of photons by more than a factor of 2. There is a similar effect for the altitude of the point on the sky. As the point moves to the horizon, the sky brightness increases. This effect is more pronounced for the bands near the blue. Finally, the sky brightness increases as the altitude of the sun increases. Perhaps more interesting is the significant color dependence of the results. Although the trends described above hold true regardless of color, the magnitude of their effect is very different. The difference between the g and y bands is about a factor of 2 difference in the point on the sky dependence and more than a factor of 10 difference in the altitude of the point on the sky. The effect of the altitude of the sun, $z_\\textrm{source}$, is more constant across color.\n\nIt is straightforward to apply the measured planar fit coefficients to an individual observation. Table~\\ref{tab:fullmoon} provides the appropriate scale factor for different moon phases and passbands. If the passband of interest varies significantly from the filters in this study, one can compute the appropriate factor from equation~\\ref{eq:fullmoon}. After this, one finds the altitude and azimuth at the site of interest at the time of the observation as well as the altitude and azimuth of the target. Three angles are then computed from these quantities: the angle between the point on the sky and the moon, the altitude of the point on the sky, and the altitude of the moon. One then computes $a + b x + c y + d y$ for the appropriate passband, where a, b, c, and d are given in Table~\\ref{tab:results}, and x corresponds to the angle between the point on the sky and the moon, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the moon.\nThis table allows for a straightforward comparison between the ratio of sky brightnesses for different colors. The ratio of fluxes in u to g, for example, are fairly flat across the sky due to the similar ratios between coefficients. On the other hand, the ratio of fluxes in u to i, depends heavily on angle to the source, with virtually no dependence on zenith angle. This indicates that the sky is much redder close to the moon than far away.\nA code that performs these steps is available at https:\/\/github.com\/mcoughlin\/skybrightness for public download. Hopefully, this will allow other researchers to easily use the data product. Required inputs are the latitude, longitude, and elevation of the site, right ascension and declination of the source, the passband of interest, and the times of observation.\n\n\n\n\\subsection{From Relative Sky Brightness to $m_5$ Variations to Optimal Scheduling}\n\nWe can use the data we have generated of {\\it relative} daytime sky brightness to generate a sky map of degradation in the point source magnitude that can be detected in the case where scattered moonlight dominates the Poisson noise. We define the {\\it sky brightness factor}, SBF, to be the ratio of the local sky to the darkest attainable sky surface brightness at that moment. In order to achieve the same SNR with varying sky backgrounds, the source must be brighter by a factor of $\\Delta m_5 = \\frac{1}{2} \\times -2.5\\,\\textrm{log}_{\\textrm{10}}(SBF)$. Because the sky brightness structure is linear in the illumination level, the sun-illuminated measurements are perfectly valid for making these $\\Delta m_5$ maps. If one region of the sky is twice as bright as another in the daytime, then for the moonlight-dominated case, replacing the sun with the moon will not change that fact. An example is shown in Figure \\ref{fig:dm5}, for the u-band. For the scheduling of LSST observations over the course of a night, a week, or a month, this\nis the format in which we think the sky brightness data is most useful. We stress that this comes directly from the daytime measurements of relative brightness, with no \nconversion needed, as long as scattered moonlight dominates the sky background. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=5.5in]{plots\/a1_alt_angle_meshgrid.pdf}\n\\caption{Variation $\\Delta m_5$ in the point source magnitude that can be detected at 5$\\sigma$ in the u-band, against spatially varying sky brightness. This contour plot shows the change in point \nsource detection threshold as a function of altitude and angular \nseparation from the moon.\nThe color bar indicates the change in $m_5$ for a fixed exposure time, in magnitudes. Maps\nsuch as this \ncan be used to optimize the sequence of LSST observations. }\n\\label{fig:dm5}\n\\end{center}\n\\end{figure}\n \n\\subsection{Prediction of Scattered Moonlight Contribution to LSST backgrounds}\n\nTable~\\ref{tab:zenithresults} presents the data analysis sequence, for sky brightness obtained at zenith with a source elevation angle of 45$^\\circ$. The table shows, for each passband, the measured photocurrent, the dark current value, the number of photoelectrons per second \nproduced in the photodiode, the geometrical factor $GF$ for scaling from solar to lunar irradiance, the \nattenuation due to the lunar phase function at $PF$ full moon ($g=2^\\circ$), the ratio $R$ of LSST pixel to photodiode etendues, the ratio of throughput times etendue for the two systems, and the\nnumber of LSST photoelectrons per pixel per second. We compute\n\n\\begin{equation}\n\\Phi_\\textrm{LSST}=\\left [ \\frac{I_\\textrm{meas}-I_\\textrm{dark}}{1.60 \\times 10^{-19} Coul} \\right ] * GF * PF * \\left [ \\frac{T_{LSST}} {T_{diodes}} \\right ] * \\left [ \\frac{(A \\Omega)_{LSST~pixel}} {(A \\Omega)_{diodes}} \\right ]\n\\end{equation}\nto obtain the expected number of photoelectrons per pixel per second we expect on the LSST focal plane, for full moon conditions, with the telescope pointed to the zenith, and a \nlunar zenith angle of 45$^\\circ$. Note that we do not include any factor for atmospheric attenuation since we wish to use the photon arrival rate on the photodiode to predict the lunar \nbackground flux on the LSST focal plane. \n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n\\hline\nBand & I$_\\textrm{meas}$ & $\\Phi_\\textrm{diode}$ & Geometry & Phase & $\\frac{T(ITL,e2v)\\,A\\,\\Omega(LSST)}{TA\\,\\Omega(Photodiode)}$ & $\\Phi_\\textrm{ITL}$ & $\\Phi_\\textrm{e2v}$ & ETC \\\\\n~ & $\\mu$A & e\/s & Factor & Factor & ~ & e\/pix\/s & e\/pix\/s &e\/pix\/s \\\\\n\\hline\nu & 1.0 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.091 & (0.61,0.45)*$1.23\\times 10^{-5}$ & 86 & 64 & 106 \\\\\ng & 3.5 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.114 & (0.62,0.66)*$1.23\\times 10^{-5}$ & 307 & 327 & 451 \\\\\nr & 1.8 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.14 & (0.65,0.67)*$1.23\\times 10^{-5}$ & 168 & 173 & 186 \\\\\ni & 1.5 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.17 & (0.50,0.50)*$1.23\\times 10^{-5}$ & 106 & 106 & 116 \\\\\nz & 2.2 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.13 & (0.67,0.66)*$1.23\\times 10^{-5}$ & 210 & 207 & - \\\\ \nzs & 0.9 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.13 & (0.67,0.66)*$1.23\\times 10^{-5}$ & 84 & 83 & 89 \\\\\ny & 1.1 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.20 & (0.40,0.40)*$1.23\\times 10^{-5}$ & 60 & 60 & 23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Zenith daytime sky brightness values, converted to expected LSST \nlunar sky backgrounds at zenith at full moon. These are all adjusted to a common angle between the point on sky and sun, altitude of the point on the sky, and altitude of sun of 45$^\\circ$. The zs band values, which are given to approximate the LSST z filter, are computed using Astrodon z minus y. There is qualitative agreement between the values calculated from the daytime measurement and the LSST exposure time calculator. The exposure time calculator uses the full sky spectrum, not just lunar part, and thus we expect the photodiode measurements to generally underestimate the exposure time calculator numbers.}\n\\label{tab:zenithresults}\n\\end{table}%\n \n\n\n \n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nMeasurements of sky brightness are important for efficient telescope scheduling and predictions for LSST. While daytime measurements are approximations to Lunar measurements, it provides high signal to noise ratio measurements in the LSST bands. We measured the fall-off in sky brightness with angle from the sun and zenith angle.\n\nThere are a number of important conclusions to draw from the measurements. The first is that there are substantial gradients in scattered sky brightness, as much as 2 magnitudes. The second is that the scattered sky brightness increases closer to the horizon, perhaps due to \nmore column density winning over extinction. The third is that when observing in bright time, there is significant benefit to point away from the moon.\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=5.5in]{plots\/delta_mag.pdf}\n\\caption{u, g, and r-band flux as a function of time for a single point on the sky using data from Table~\\ref{tab:zenithresults}. Code to produce this plot is available at https:\/\/github.com\/mcoughlin\/skybrightness for public download. This shows in general the differences in flux for a single observation throughout the night.}\n\\label{fig:deltam}\n\\end{center}\n\\end{figure} \n \nAs motivation for future work, figure~\\ref{fig:deltam} shows u, g, and r-band flux as a function of time for a single point on the sky using the measurements described in this paper.\nWe can use these measurements to prioritize observations throughout a given night.\nIn the future, we intend to improve on these measurements by designing the apparatus to take simultaneous measurements. \nWith such an apparatus, we will be able to, for example, measure the color of clouds. \nWe will also be able to take significantly more observations, allowing for refinement of this model, continuing to make it more useful for observers and those exploring scheduling strategies.\n\n\\section{Acknowledgments}\nMC was supported by the National Science Foundation Graduate Research Fellowship\nProgram, under NSF grant number DGE 1144152. CWS is grateful to the DOE Office \nof Science for their support under award DE-SC0007881. Thanks also to Prof. Gary Swensen of Univ. of Illinois for hospitality in the ALO building at Pachon.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\nLet $(X_1, X_2,\\dots, X_n)$ be $n$ independent random points uniformly distributed on the square $\\cc01^2$. \nThe semi-discrete random matching problem concerns the study of the properties of the optimal \ncoupling (with respect to a certain cost) of these $n$ points with the Lebesgue measure \n$\\restricts{\\Leb^2}{\\cc01^2}$. \n\nMore precisely, denoting $\\mu^n \\defeq \\frac1n \\sum_{i=1}^n\\delta_{X_i}$ the empirical measure and \n$\\m\\defeq\\restricts{\\Leb^2}{\\cc01^2}$, we want to investigate the optimal transport from \n$\\m$ to $\\mu^n$.\n\nThe ultimate goal is understanding both the distribution of the random variable associated to the optimal \ntransport cost and the properties of the (random) optimal map. In the present paper we will show that\nthe optimal transport map can be well-approximated by the identity plus the gradient of the solution \nof a Poisson problem. In the large literature devoted to the matching problem,\nwe believe that (except for the 1-dimensional case) this is one of the few results describing\nthe behavior of the optimal map, and not only of the transport cost, see also \n\\cite{Goldman2018} in connection with the behavior of the optimal transport map in the Lebesgue-to-Poisson problem \non large scales. \n\nBefore going on, let us briefly recall the definitions of optimal transport and Wasserstein distance. \nWe suggest the monographs \\cite{Villani08,Santambrogio15} for an introduction to the topic.\n\\begin{definition}[Wasserstein distance]\n Let $(X, d)$ be a compact metric space and let $\\mu,\\nu\\in\\prob(X)$ be probability measures.\n Given $p\\in\\co{1}{\\infty}$, we define the $p$-Wasserstein distance between $\\mu$ and $\\nu$ as\n \\begin{equation}\\label{eq:defWp}\n W_p^p(\\mu, \\nu) \\defeq \\inf_{\\gamma\\in\\Gamma(\\mu, \\nu)} \\int_{X\\times X} d^p(x, y) \\de\\gamma(x, y) \\comma\n \\end{equation}\n where $\\Gamma(\\mu, \\nu)$ is the set of all $\\gamma\\in\\prob(X\\times X)$ such that the projections $\\pi_i$, $i=1,2$, on the \n two factors are $\\mu$ and $\\nu$, that is $(\\pi_1)_\\#\\gamma = \\mu$ and $(\\pi_2)_\\#\\gamma=\\nu$.\n\\end{definition}\n\\begin{remark}\n The infimum in the previous definition is always attained (\\cite[Theorem 1.4]{Santambrogio15}).\n \n Moreover, if $(X, d)$ is a Riemannian manifold and $\\mu\\ll\\m$, where $\\m$ is the volume measure of \n the manifold, the Wasserstein distance is realized by a map (\\cite{McCann01}). \n Namely, the infimum \\cref{eq:defWp} is attained and the unique minimizer is induced by a Borel map $T:M\\to M$, \n so that $T_\\#\\mu = \\nu$ and \n \\begin{equation*}\n W_p^p(\\mu, \\nu) = \\int_M d^p(x, T(x)) \\de\\mu(x) \\fullstop\n \\end{equation*}\n\\end{remark}\n\nEven though the square is a fundamental example, the random matching problem makes perfect sense even in \nmore general spaces (changing the reference measure $\\m$ accordingly).\nHistorically, in the combinatorial literature\\footnote{In the combinatorial literature the problem considered\nwas the bipartite matching problem, in which two independent random point clouds have to be matched. The \nsemi-discrete matching and the bipartite matching are tightly linked and, given that we will consider only\nthe former, we are going to talk about the combinatorial literature as if it were considering the semi-discrete\nmatching.}, the most common ambient space was $\\cc01^d$ for some $d\\ge 1$ and the aspect of the problem that \nattracted more attention was estimating the expected value of the $W_1$ cost. \nIn the papers \\cite{Ajtai1984,Talagrand1992,Dobric1995,Ledoux2017} (and possibly in other ones) \nthe problem was solved in all dimensions and for all $1\\le p < \\infty$, obtaining the growth \nestimates\\footnote{The notation $f(n)\\approx g(n)$ means that there exists a positive constant $C>0$ such \nthat $C^{-1}g(n) \\le f(n)\\le C g(n)$ for every $n$.}\n\\begin{equation*}\n \\E{W_p^p(\\m, \\mu^n)} \\approx \n \\begin{cases}\n n^{-\\frac p2} &\\text{ if $d=1$,}\\\\\n \\left(\\frac{\\log(n)}n\\right)^{\\frac p2} &\\text{ if $d=2$,}\\\\\n n^{-\\frac pd} &\\text{ if $d\\ge 3$.}\n \\end{cases} \n\\end{equation*}\nAs might be clear from the presence of a logarithm, the matching problem exhibits some unexpected behavior\nin dimension $2$. \n\nSee the introductions of \\cite{ambrosio-glaudo2018,Ledoux2017} or \\cite[Chapter 4, 14, 15]{Talagrand14}\nfor a more in-depth description of the history of the problem.\n\nNowadays the topic is active again (\\cite{holden2018,Talagrand2018,Goldman2018,Ledoux2017,ambrosio-glaudo2018,Ledoux18,Ledoux19}),\nalso as a consequence of \\cite{Ambrosio-Stra-Trevisan2018}, in which the authors, \nfollowing an ansatz suggested in \\cite{CaraccioloEtAl2014}, manage to obtain the leading term of the \nasymptotic expansion of the expected matching cost in dimension $2$ with respect to the quadratic \ndistance\\footnote{The notation $f(n)\\sim g(n)$ means that $\\frac{f(n)}{g(n)}\\to 1$ when $n\\to\\infty$.}:\n\\begin{equation}\\label{eq:limit_value}\n \\E{W_2^2(\\m, \\mu^n)} \\sim \\frac{\\log(n)}{4\\pi n} \\fullstop\n\\end{equation}\nThe approach is far from being combinatorial, indeed it relies on a first-order approximation of the \nWasserstein distance with the $H^{-1}$ negative Sobolev norm. Their proof works on any closed compact \n$2$-dimensional manifold.\n\nGiven that we will build upon it, let us give a brief sketch of the approach. \nWhat we are going to describe is simpler than the original approach of \\cite{Ambrosio-Stra-Trevisan2018} and\ncan be found in full details in \\cite{ambrosio-glaudo2018}. For simplicity we will assume to work on the \nsquare.\n\nLet $T^n$ be the optimal map from $\\m$ to $\\mu^n$, whose existence is ensured by Brenier's Theorem \n(see \\cite{Brenier91}). Still by Brenier's Theorem, we know that $T^n = \\id + \\nabla \\tilde f^n$, where\n$\\id$ is the identity map and $\\tilde f^n:\\cc01^2\\to M$ is a convex function. \nWith high probability $\\mu^n$ is well-spread on the square, thus we expect $\\nabla \\tilde f^n$ to be \n\\emph{very small}.\nWe know $(T^n)_\\#\\m=\\mu^n$ and we would like to apply the change of variable formula to deduce something \non the Hessian of $\\tilde f^n$. \nThe issue is that the singularity of $\\mu^n$ prevents a direct application of the change of variable \nformula. \nAnyhow, proceeding formally we obtain $\\det(\\id+\\nabla^2 \\tilde f^n)^{-1}=\\mu^n$. \nGoing on with the formal computation, if we consider only the first order term of \nthe left hand side, the previous identity simplifies to\n\\begin{equation*}\n -\\lapl \\tilde f^n \\approx \\mu^n-1 \\fullstop\n\\end{equation*}\nSomewhat unexpectedly, this last equation makes perfect sense. \nTherefore we might claim that if we define $f^n:\\cc01^2\\to\\R$ as the solution of $-\\lapl f^n=\\mu^n-1$\n(with null Neumann boundary condition), then $T^n$ is well-approximated by $\\id+\\nabla f^n$ and \nfurthermore the transport cost is well-approximated by $\\int \\abs{\\nabla f^n}^2\\de\\m$.\n\nThis conjecture is appealing, but false, if taken literally. Indeed, it is very easy to check that the integral\n$\\int \\abs{\\nabla f^n}^2\\de\\m$ diverges. \n\nThe ingredient that fixes this issue is a regularization argument. \nMore precisely, let $\\mu^{n,t}\\defeq P_t^*\\mu^n$ be the evolution at a certain small time $t>0$ of the \nempirical measure through the heat semigroup (see \\cite[Chapter 6]{Chavel84}).\nIf we repeat the ansatz with $\\mu^n$ replaced by $\\mu^{n,t}$ we obtain a function $f^{n,t}:\\cc01^2\\to\\R$ \nthat solves \n\\begin{equation*}\n -\\lapl f^{n,t} = \\mu^{n,t}-1\n\\end{equation*}\nwith null Neumann boundary conditions. Let us remark that in fact $f^{n,t}=P_t f^n$.\n\nOnce again, we can hope that $\\id+\\nabla f^{n,t}$ approximates very well $T^n$ and furthermore that the transport\ncost from $\\m$ to $\\mu^n$ is well-approximated by $\\int \\abs{\\nabla f^{n,t}}^2\\de\\m$.\n\nThis time the predictions are sound.\nChoosing carefully the time $t=t(n)$, we can show that, with high probability, the map $\\id+\\nabla f^{n,t}$ is\noptimal from $\\m$ to $\\left(\\id+\\nabla f^{n,t}\\right)_\\#\\m$ and the Dirichlet energy of $f^{n,t}$ approximate\nvery well $W_2^2(\\m, \\mu^n)$.\nOnly one part of the conjecture is left unproven by \\cite{ambrosio-glaudo2018}: is it true that \n$\\id+\\nabla f^{n,t}$ approximates, in some adequate sense, the optimal map $T^n$?\nThe goal of the present paper is to answer positively this question.\n\nWe are going to prove the following.\n\\begin{theorem}\\label{thm:main_theorem}\n Let $(M,\\metric)$ be a $2$-dimensional closed compact Riemannian manifold (or the square $\\cc01^2$) whose\n volume measure $\\m$ is a probability. We will denote with $d:M\\times M\\to\\co0\\infty$ the Riemannian \n distance on $M$.\n \n Given $n\\in\\N$, let $X_1, X_2,\\dots, X_n$ be $n$ independent random points $\\m$-uniformly \n distributed on $M$. \n Let us denote $\\mu^n\\defeq \\frac1n \\sum_i \\delta_{X_i}$ the empirical measure associated to the \n random point cloud and let $T^n$ be the optimal transport map from $\\m$ to $\\mu^n$.\n \n For a fixed time $t>0$, let $\\mu^{n,t}\\defeq P_t^*\\mu^n\\in\\prob(M)$ and let $f^{n,t}:M\\to\\R$ be the unique\n null-mean solution\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the null Neumann boundary\n conditions.} of the Poisson problem $-\\lapl f^{n,t}=\\mu^{n,t}-1$.\n \n If we set $t=t(n)=\\frac{\\log(n)^4}n$, on average $T^n$ is very close to $\\exp(\\nabla f^{n,t})$ in the \n $L^2$-norm, that is\n \\begin{equation}\\label{eq:main-quantitative}\n \\frac{\\E{\\int_M d^2(T^n, \\exp(\\nabla f^{n,t}))\\de\\m}}{\\frac{\\log(n)}{n}} \\ll \\sqrt{\\frac{\\log \\left(\\log (n)\\right)}{ \\log(n) }} \\fullstop\n \\end{equation}\n In particular,\n \\begin{equation*}\n \\lim_{n\\to\\infty}\\frac{\\E{\\int_M d^2(T^n, \\exp(\\nabla f^{n,t}))\\de\\m}}\n {\\E{\\int_M d^2(T^n, \\id)\\de\\m}} \n = 0 \\fullstop\n \\end{equation*}\n\\end{theorem}\n\\begin{remark}\n To handle the case of the square $M=\\cc01^2$ some care is required. Indeed the presence of boundary makes\n things more delicate. This is the reason why only the square is considered in the theorem and not any\n $2$-dimensional compact manifold with boundary. \n \n See \\cite[Subsection 2.1 and Remark 3.10]{ambrosio-glaudo2018} for some further details on this matter.\n\\end{remark}\n\n\\begin{remark}\\label{rem:distance-tangent}\nBy McCann's Theorem \\cite{McCann01} we can write $T^n = \\exp(\\nabla f^n)$, hence a natural \nquestion is if \\cref{eq:main-quantitative} holds with $|\\nabla (f^n - f^{n,t})|$ in place of \n$d(T^n, \\exp(\\nabla f^{n,t}))$. Using the fact that the exponential map restricted to a \nsufficiently small neighbourhood of the null vector field is a global diffeomorphism with its \nimage, it would be sufficient to show that, for every \n$\\varepsilon>0$, $\\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\ll \\log(n)\/n$, as $n \\to \\infty$. \nWe will prove this estimate in \\cref{prop:linf_is_small}, that provides the desired approximation at\nthe level of the gradients\n\\begin{equation}\\label{eq:main_gradients}\n \\lim_{n\\to\\infty} \n \\frac{\\E{\\norm{\\nabla f^n-\\nabla f^{n,t}}_{L^2(M)}^2}}\n {\\E{\\norm{\\nabla f^n}_{L^2(M)}^2}} = 0\\fullstop \n\\end{equation}\n\\end{remark}\n\nThe strategy of the proof is to show that the information that we already have on $\\exp(\\nabla f^{n,t})$ \n(namely that it is an optimal map between $\\m$ and some measure $\\hat\\mu^{n,t}$ that is very close to \n$\\mu^{n,t}$) is enough to deduce that it must be near to the optimal map $T^n$. \n\nAs part of the strategy of proof, we obtain, in \\cref{sec:stability}, a new stability result for the optimal transport \nmap on a general compact Riemannian manifold (not only of dimension $2$). \nThis is the natural generalization to Riemannian manifolds of \\cite{gigli2011}.\nThe said stability result follows rather easily from the study of the short-time behavior of the Hopf-Lax \nsemigroup we perform in \\cref{sec:hopflax}. \nThe Hopf-Lax semigroup comes up in our investigation as, when $t=1$, it becomes the operator of\n$c$-conjugation and thus produce the second Kantorovich potential once the first is known (see \n\\cite[Section 1.2]{Santambrogio15} for the theory of Kantorovich potentials and $c$-conjugation).\n\nThe main theorem is established in \\cref{sec:random_matching}.\n\\vspace{2mm}\n\n\\noindent {\\textit{Acknowledgments. } F. Glaudo has received funding from the European Research Council under the \nGrant Agreement No 721675. L. Ambrosio acknowledges the support of the MIUR PRIN 2015 project.\n\n\\subsection{Notation for constants}\nWe will use the letters $c$ and $C$ to denote constants, whose dependencies are denoted by $c=c(A, B,\\dots)$. \nThe value of such constants can change from one time to the other.\n\nMoreover we will frequently use the notation $A\\lesssim B$ to hide a constant that depends only on the\nambient manifold $M$. This expression means that there exists a constant $C=C(M)$ such that $A\\le C\\cdot B$.\n\n\n\n\n\\section{Short-time behavior of the Hopf-Lax semigroup with datum in \\texorpdfstring{$C^{1,1}$}{C(1,1)}}\n\\label{sec:hopflax}\nLet us begin recalling the definition of the Hopf-Lax semigroup (also called Hamilton-Jacobi semigroup).\n\\begin{definition}[Hopf-Lax semigroup]\n Let $(X, d)$ be a compact length space\\footnote{A metric space is a length space if the distance between\n any two points is the infimum of the length of the curves between the two points. Let us remark that for\n the definition we need neither the compactness nor the length property of $X$, but without these \n assumptions many of the properties of the Hopf-Lax semigroup fail (first of all the fact that it is \n a semigroup).}.\n For any function $f\\in C(X)$ and any $t\\geq 0$, let $Q_t f:X\\to\\R$ be defined by\n \\begin{equation*}\n Q_t f(y) = \\min_{x\\in X} \\frac1{2t} d^2(x,y) + f(x) \\quad (t>0),\\qquad Q_0f=f\\fullstop\n \\end{equation*}\n\\end{definition}\nWithout additional assumptions on $X$ or $f$ it is already possible to deduce many properties of the Hopf-Lax\nsemigroup. Let us give a very short summary of the most important ones.\n\\begin{itemize}\n \\item When $t\\to 0$ the functions $Q_t f$ converge uniformly to $f$.\n \\item The Hopf-Lax semigroup is indeed a semigroup, that is $Q_{s+t}f = Q_sQ_t f$ for any $s,\\, t \\geq 0$.\n \\item In a \\emph{suitable weak sense},the Hamilton-Jacobi equation\n \\begin{equation*}\n \\frac{\\de}{\\de t} Q_t f + \\frac12\\abs{\\nabla Q_t f}^2 = 0 \n \\end{equation*}\n holds.\n Let us emphasize that the mentioned equation does not make sense if we don't give an appropriate definition\n of norm of the gradient as we are working in a metric setting.\n\\end{itemize}\nSee \\cite{Lott-Villani07}, in particular Theorem~2.5, for a detailed proof of the mentioned\nproperties. \n\nThere is a vast literature investigating the regularity properties of the Hopf-Lax semigroup and its \nconnection with the Hamilton-Jacobi equation, in particular that it is the unique solution in the viscosity\nsense (see for instance \\cite{lions1982,benton1977,bardi2008}). \nNonetheless we could not find a complete reference for the short-time behavior of the Hopf-Lax semigroup on \na Riemannian manifold (as the majority of the results are stated on the Euclidean space) with a relatively \nregular initial datum (namely $C^{1,1}$). This is exactly the topic of this section. \n\nWhat we are going to show, apart from \\cref{it:hopflax_convexity}, is not new. For instance,\nin \\cite[Section 5]{fathi2003}, the author proves the validity of the method of characteristics in \na way very similar to ours. In that paper more general Lagrangians are considered and as a consequence \nthe proofs are more involved and require much more geometric tools and notation.\n\nFor us, the ambient space is a compact Riemannian manifold $(M, \\metric)$ and the function \n$f\\in C^{1,1}(M)$ is differentiable with Lipschitz continuous gradient. Moreover, either $M$ is closed\nor it is the square $\\cc01^2$. For a general\nmanifold with boundary the results are false, the square is special because its boundary is piecewise geodesic.\nHandling all manifolds with totally geodesic boundary would be possible, but would require some additional \ncare. In order to simplify the exposition we decided to state the results only for the square.\nThroughout this section we will often use implicitly that a Lipschitz continuous function is differentiable\nalmost everywhere (see \\cite[Theorem 3.2]{Evans-Gariepy}).\n\nWe will show that, up to a small time that depends on the $C^{1,1}$-norm of $f$, the Hopf-Lax semigroup \nis \\emph{as good as one might hope}.\nWe will describe explicitly the minimizer $x=x_t(y)$ of the variational problem that \ndefines $Q_t f(y)$ deducing some \\emph{explicit} formulas for $Q_tf$ and its gradient and we will\nshow that $Q_t f$ solves the Hamilton-Jacobi equation in the classical sense.\nFinally we will be able to control the $C^{1,1}$-norm of $Q_t f$ and the $C^{0,1}$-norm of $Q_t f-f$. \n\nHow can we achieve these results for short times when $f\\in C^{1,1}$? \nThe main ingredient is the possibility to identify the minimizer $x=x_t(y)$ in the definition of $Q_t f(y)$.\nGiven $x\\in M$, let $\\gamma:\\co{0}{\\infty}\\to M$ be the unique geodesic with $\\gamma(0) = 0$ and \n$\\gamma'(0)=\\nabla f(x)$. If $y=\\gamma(t)$, then the minimizer in the definition of $Q_t f(y)$ is exactly $x$.\nThis approach is exactly the method of characteristics when applied on a Riemannian manifold (\\emph{straight \nlines on a manifold are geodesics}).\n\nLet us begin with a technical lemma. \n\\begin{lemma}\\label{lem:exp_is_diffeo}\n Let $(M, \\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$).\n\n There exists a constant $c=c(M)$ such that the following statement holds.\n Let $X\\in\\chi(M)$ be a Lipschitz continuous vector \n field\\footnote{If $M=\\cc01^2$ we ask also that $X$ is tangent to the boundary.}\n with $\\norm{X}_\\infty\\le c$ and $\\norm{\\nabla X}_\\infty\\le c$ and, for any $0\\le t\\le 1$, let \n $\\varphi_t:M\\to M$ be the map defined as $\\varphi_t(x) \\defeq \\exp(tX(x))$, where $\\exp:TM\\to M$ denotes the \n exponential map.\n For any $0\\le t\\le 1$, the map $\\varphi_t$ is a homeomorphism \n such that $\\Lip(\\varphi_t)$, $\\Lip(\\varphi_t^{-1}) \\le 2$ and the vector field $X_t\\in\\chi(M)$ defined\n as\n \\begin{equation*}\n X_t \\defeq \\frac{\\partial \\varphi_s}{\\partial s}\\Big|_{s=t}\n \\end{equation*}\n is Lipschitz continuous with $\\norm{\\nabla X_t}_{\\infty}\\lesssim\\norm{\\nabla X}_{\\infty}$.\n\\end{lemma}\n\\begin{proof}\n We will give only a sketch of the proof of the first part of the statement as the argument is well-known.\n\n Let us begin by proving the result when $M$ is closed (in particular we exclude only $M=\\cc01^2$).\n \n We can deduce the first part of the statement from the fact that $\\varphi=\\varphi_1$ is injective and \n locally (i.e. on sufficiently small balls) it is a bi-Lipschitz transformation with its image.\n \n Working in a suitably chosen finite atlas (whose existence follows from the compactness of $M$), the fact \n that $\\varphi$ is a bi-Lipschitz diffeomorphism is a consequence of the following very \n well-known lemma about perturbations of the identity (see \\cite[Theorem 9.24]{rudin1976} or \n \\cite[Theorem 5.3]{fathi2003}).\n If $T:\\Omega\\subseteq \\R^d\\to\\R^d$ is such that $T-\\id$ is $L$-Lipschitz with $L<1$, then $T$ is locally\n invertible and $\\Lip(T) \\le 1+L,\\ \\Lip(T^{-1}) \\le (1-L)^{-1}$.\n \n The global injectivity follows directly from the fact that it is locally bi-Lipschitz. Indeed if \n $\\varphi(x_1)=\\varphi(x_2)$ then $d(x_1, x_2) \\le 2\\norm{X}_\\infty$ and therefore we can exploit the local\n injectivity of $\\varphi$.\n \n When $M=\\cc01^2$ we need only a simple additional remark. Given that $X$ is tangent to the boundary, the map \n $\\varphi$ is a homeomorphism of the boundary. As a consequence of this fact, it is not\n difficult to prove (by injectivity) that the image of the interior of the square is mapped \n by $\\varphi$ in itself. \n From here on we can simply mimic the proof described above for closed manifolds and achieve the\n result also for the case of the square.\n \n We move our attention to the second part of the statement.\n By a simple homogeneity argument, it is sufficient to prove that\n $\\norm{\\nabla X_t}_{\\infty}\\lesssim 1$.\n \n Once again we work in chart. Let $\\Omega\\subseteq\\R^d$ be the domain of the chart. \n As usual, $X_t$ can be understood as a vector field on $\\Omega$ and $\\varphi_t$ as a map from \n $\\Omega'\\Subset\\Omega$ into $\\Omega$.\n Choosing the chart appropriately, we can assume that the Euclidean distance is bi-Lipschitz equivalent\n to the distance induced by the metric $\\metric$.\n \n The Lipschitz continuity of $X_t$ with respect to the metric $\\metric$ is equivalent to proving that, for \n any $x,y\\in\\Omega$, it holds\n \\begin{equation*}\n \\abs{X_t(x)-X_t(y)} \\lesssim \\abs{x-y} \\comma\n \\end{equation*}\n where all the absolute values are with respect to the standard Euclidean norm. \n Since $\\varphi_t$ is surjective, it is sufficient to prove that, for any $x,y\\in\\Omega'$, it holds\n \\begin{equation}\\label{eq:exp_diffeotmp1}\n \\abs{X_t(\\varphi_t(x))-X_t(\\varphi_t(y))} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \\fullstop\n \\end{equation}\n \n Given that $\\varphi_t^{-1}$ is Lipschitz, we already know\n \\begin{equation}\\label{eq:exp_diffeotmp2}\n \\abs{x-y} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \n \\quad\\text{and}\\quad\n \\abs{X(x)-X(y)} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \\fullstop\n \\end{equation}\n Let $\\gamma_x:\\cc01\\to\\Omega$ be the unique geodesic, with respect to $\\metric$, such that $\\gamma_x(0)=x$\n and $\\gamma_x'(0)=X(x)$. Let $\\gamma_y:\\cc01\\to\\Omega$ be defined analogously. By definition, it holds\n \\begin{equation}\\label{eq:exp_diffeotmp3}\n X_t(\\varphi_t(x)) = \\gamma_x'(t)\n \\quad\\text{and}\\quad\n X_t(\\varphi_t(y)) = \\gamma_y'(t) \\fullstop\n \\end{equation}\n Taking into account \\cref{eq:exp_diffeotmp1}, \\cref{eq:exp_diffeotmp2} and \\cref{eq:exp_diffeotmp3}, the \n Lipschitz continuity of $X_t$ would follow from the inequality\n \\begin{equation}\\label{eq:exp_diffeotmp4}\n \\abs{\\gamma_x'(t)-\\gamma_y'(t)}\n \\lesssim \\abs{\\gamma_x(0)-\\gamma_y(0)} + \\abs{\\gamma_x'(0)-\\gamma_y'(0)}\n \\fullstop\n \\end{equation}\n The curves $\\gamma_x,\\gamma_y$ are geodesics, hence the vectors $(\\gamma_x, \\gamma_x')$ and \n $(\\gamma_y,\\gamma_y')$ solve the same autonomous ordinary differential equation with different initial data.\n Hence \\cref{eq:exp_diffeotmp4} follows from the well-known Lipschitz dependence of the solution \n from the initial data (see \\cite[Theorem 2.6]{teschl2012}) and therefore the proof is concluded.\n\\end{proof}\n\nWe can now state and prove the main theorem of this section. \nThe technically demanding part of these notes is entirely enclosed in the following theorem.\n\\begin{theorem}\\label{thm:hopflax_properties}\n Let $(M, \\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$).\n \n Let $f\\in C^{1,1}(M)$ be a scalar function\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the \n null Neumann boundary conditions.} and, for any positive time $t>0$, let us define the map \n $\\varphi_t:M\\to M$ as $\\varphi_t(x) \\defeq \\exp(t\\nabla f(x))$.\n \n There exists a constant $c=c(M)$ such that the following properties hold for any time \n $0\\le t\\le c \\left(\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty\\right)^{-1}$:\n \\begin{enumerate}[ref={(\\arabic*)}]\n \\item \\label{it:varphi_t_diffeo}\n The map $\\varphi_t$ is a bi-Lipschitz homeomorphism such that \n $\\Lip(\\varphi_t),\\, \\Lip(\\varphi_t^{-1}) \\leq 2$.\n \\item \\label{it:hopflax_explicit} For any $y\\in M$, it holds\n \\begin{equation*}\n Q_t f(y) = \\frac1{2t} d^2(\\varphi_t^{-1}(y), y) + f(\\varphi_t^{-1}(y)) \\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_convexity}\n For any $y,\\,y'\\in M$, one has the (strict-convexity-like) estimate\n \\begin{equation*}\n \\frac{d^2(y, y')}t \\lesssim Q_tf(y)-Q_tf(y')\n +\\frac1{2t}\\left[d^2(\\varphi_t^{-1}(y), y') - d^2(\\varphi_t^{-1}(y), y)\\right]\\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_regularity}\n The function $Q_tf$ is Lipschitz continuous in time and $C^{1,1}(M)$ in space. In particular we have\n $\\norm{\\partial_t Q_t f}_{\\infty}\\le \\norm{\\nabla f}_{\\infty}$ and\n $\\norm{\\nabla^2Q_tf}_{\\infty} \\lesssim \\norm{\\nabla^2 f}_{\\infty}$.\n \\item \\label{it:hamilton_jacobi} \n The function $Q_tf$ is a classical solution of the Hamilton-Jacobi equation\n \\begin{equation*}\n \\frac{\\de}{\\de t} Q_t f + \\frac12 \\abs{\\nabla Q_t f}^2 = 0 \\fullstop\n \\end{equation*}\n \\item \\label{it:gradient_conservation} \n For any $x\\in M$, if $\\gamma:\\cc01\\to M$ is the geodesic such that $\\gamma(0)=x$ and \n $\\gamma'(0) = \\nabla f(x)$, then it holds\n \\begin{equation*}\n Q_tf(\\gamma(t)) = f(x) + \\frac t2 \\abs{\\nabla f}^2(x) \\quad\\text{ and }\\quad\n \\nabla Q_t f(\\gamma(t)) = \\gamma'(t) \\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_lip}\n One has\n \\begin{equation*}\n \\Lip(Q_tf-f) \\le t\\norm{\\nabla f}_\\infty\\cdot\\norm{\\nabla^2 f}_\\infty \\fullstop\n \\end{equation*}\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n Thanks to the following homogeneity, for any $t>0$ and $\\lambda>0$, of the Hopf-Lax semigroup\n \\begin{equation*}\n Q_t(\\lambda f)(y) = \\lambda Q_{\\lambda t} f(y) \\comma\n \\end{equation*}\n we can assume without loss of generality that $\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty \\le c$ \n and prove that the statements hold up to time $1$.\n Thus, we will implicitly assume that the time variable satisfies $0\\le t\\le 1$.\n We will choose the value of the constant $c$ during the proof, it should be clear that all constraints \n we impose depend only on the manifold $M$ and not on the function $f$.\n\n The statement of \\cref{it:varphi_t_diffeo} follows from \\cref{lem:exp_is_diffeo}.\n \n To prove \\cref{it:hopflax_explicit} we need some preliminary observations. \n If $c=c(M)$ is sufficiently small (so that the constraint on $f$ is sufficiently strong), thanks to \n the compactness of $M$ we can find a radius $r=r(M)>0$ such that:\n \\begin{enumerate}[label=(\\alph*)]\n \\item If $p,\\,q\\in M$ satisfy $d(p,q)\\le r$ then\n \\begin{equation*}\n \\nabla^2 d^2(\\emptyparam,p)(q) \\ge \\frac12 \\metric \\fullstop\n \\end{equation*}\n \\item For any $y\\in M$, to compute $Q_tf(y)$ it is sufficient to minimize on $B(y,r)$:\n \\begin{equation*}\n Q_tf(y) = \\inf_{x\\in B(y, r)} \\frac1{2t}d^2(x, y)+f(x) \\fullstop\n \\end{equation*}\n \\item For any $y\\in M$ it holds the inequality $d(y, \\varphi_t^{-1}(y)) \\le r$. \n In particular we can assume that $\\varphi_t^{-1}(y)$ is not in the cut-locus of $y$.\n \\item For any $y\\in M$ it holds the identity\n \\begin{equation*}\n \\nabla \\left(\\frac1{2t}d^2(\\emptyparam, y) + f(\\emptyparam)\\right)(\\varphi_t^{-1}(y)) = 0 \\fullstop\n \\end{equation*}\n This identity can be shown computing the gradient of the distance from $y$ squared, since we know\n that $y=\\exp(t\\nabla f(x))$ where $x=\\varphi_t^{-1}(y)$. Indeed, given that $x$ does not belong to the \n cut-locus of $y$, we know\n \\begin{equation*}\n \\nabla \\left(\\frac12 d^2(\\emptyparam, y)\\right)(x) = -t\\nabla f(x)\n \\end{equation*}\n and the desired identity follows.\n \\end{enumerate}\n With these observations at our disposal, the proof of \\cref{it:hopflax_explicit} is straight-forward.\n Given a time $0\\le t\\le 1$ and a point $y\\in M$, let us consider the function $w_{t,y}:M\\to\\R$ defined\n as\n \\begin{equation*}\n w_{t,y}(x) = \\frac1{2t}d^2(x,y) + f(x) \\fullstop\n \\end{equation*}\n We know that $Q_tf(y) = \\min_{x\\in B(y, r)} w_{t,y}(x)$. Moreover $\\nabla w_{t,y}(\\varphi_t^{-1}(y))=0$ and,\n if the constraint on $\\norm{\\nabla^2 f}_\\infty$ is sufficiently small, we also know \n $\\nabla^2 w_{t,y}\\ge \\frac1{3t}\\metric$ in $B(y, r)$. \n Hence, by convexity, we deduce that $\\varphi_t^{-1}(y)$ is the global minimum point of \n $w_{t,y}$ and \\cref{it:hopflax_explicit} follows.\n \n Let us now move to the proof of \\cref{it:hopflax_convexity}. \n Let $x,\\, x'\\in M$ be such that $\\varphi_t(x) = y$ and $\\varphi_t(x')=y'$. \n Applying \\cref{it:hopflax_explicit} and recalling that $\\varphi_t$ is a bi-Lipschitz \n diffeomorphism, we can see that the inequality we want to prove is equivalent to\n \\begin{equation*}\n \\frac1t d^2(x, x') \\lesssim f(x)-f(x') + \\frac 1{2t}\\left(d^2(x, y')-d^2(x', y') \\right)\n \\end{equation*}\n and, using the same notation as above, this becomes\n \\begin{equation*}\n \\frac1t d^2(x, x') \\lesssim w_{t, y'}(x) - w_{t, y'}(x') \\fullstop\n \\end{equation*}\n The latter inequality follows from the strict convexity of $w_{t,y'}$ that we have already shown \n while proving \\cref{it:hopflax_explicit}.\n\n Showing from scratch that $Q_tf$ solves the Hamilton-Jacobi equation would not be hard, but for this \n we refer to \\cite[Theorem 2.5, viii]{Lott-Villani07}, where the authors show that $Q_tf$ is \n a \\emph{suitably weak} solution of the Hamilton-Jacobi equation. \n From their statement, we can deduce that if $Q_tf$ is differentiable at $x\\in M$, then\n \\begin{equation}\\label{eq:hamilton_jacobi_ae}\n \\frac{\\de}{\\de t}Q_tf(x) + \\abs{\\nabla Q_tf(x)}^2 = 0 \\fullstop\n \\end{equation}\n Since we will show that $Q_tf$ is $C^{1,1}(M)$, the validity of \n \\cref{it:hopflax_regularity,it:hamilton_jacobi} is a consequence of \\cref{eq:hamilton_jacobi_ae}.\n \n The first part of \\cref{it:gradient_conservation}, namely \n $Q_tf(\\gamma(t)) = f(x) + \\frac t2\\abs{\\nabla f}^2(x)$, is implied by \\cref{it:hopflax_explicit}. \n To obtain the identity involving the gradient, let us differentiate the previous equality with respect to the\n time variable. If $Q_t f$ is differentiable at $\\gamma(t)$, it holds\n \\begin{equation}\\label{eq:almost_gradient_conservation}\n \\frac{\\de}{\\de t} (Q_t f)(\\gamma(t)) + \\scalprod{\\nabla Q_t f(\\gamma(t))}{\\gamma'(t)} = \n \\frac{\\de}{\\de t} (Q_tf(\\gamma(t))) = \\frac12\\abs{\\nabla f}^2(x)\\fullstop\n \\end{equation}\n Applying \\cref{it:hamilton_jacobi} and the fact that $\\abs{\\gamma'(t)} = \\abs{\\nabla f}(x)$, from\n \\cref{eq:almost_gradient_conservation} we can deduce\n \\begin{equation}\\label{eq:hopflax_tmp}\n -\\frac12 \\abs{\\nabla Q_tf}^2(\\gamma(t)) + \\scalprod{\\nabla Q_t f(\\gamma(t))}{\\gamma'(t)} \n = \\frac12\\abs{\\gamma'}^2(x) \\iff \\abs{\\nabla Q_tf(\\gamma(t)) - \\gamma'(t)}^2 = 0 \\fullstop\n \\end{equation}\n This does not imply directly \\cref{it:gradient_conservation} since we have shown the identity only if\n $Q_t f$ is differentiable at $\\gamma(t)$.\n As a byproduct of \\cref{it:hopflax_explicit}, we know that $Q_t f$ is Lipschitz continuous and \n therefore, from \\cref{eq:hopflax_tmp}, we can deduce that, fixed $t$, for almost every $x\\in M$ it holds\n \\begin{equation*}\n \\nabla Q_t f(\\varphi_t(x)) = \\frac{\\partial \\varphi_s(x)}{\\partial s}\\Big|_{s=t} \\fullstop\n \\end{equation*}\n Since the right-hand side is Lipschitz continuous (see \\cref{lem:exp_is_diffeo}) it follows that \n $Q_tf\\in C^{1,1}(M)$ and, as anticipated, this concludes the proofs of \n \\cref{it:hopflax_regularity},\\cref{it:hamilton_jacobi} and \\cref{it:gradient_conservation}.\n \n Finally let us tackle \\cref{it:hopflax_lip}.\n Given $y\\in M$, let $x=\\varphi_t^{-1}(y)$. Thanks to \\cref{it:gradient_conservation}, if we consider \n the geodesic $\\gamma:\\cc01\\to M$ such that $\\gamma(0)=x$ and $\\gamma'(0)=\\nabla f(x)$, we know that \n $\\gamma(t)=y$ and $\\gamma'(t)=\\nabla Q_tf(y)$.\n \n Thus we have\n \\begin{equation*}\n \\abs{\\nabla f(y) -\\nabla Q_tf(y)} \n \\le \\int_0^t \\abs{\\nabla_{\\gamma'}\\left(\\nabla f(\\gamma)-\\gamma'\\right)}\\de s\n \\le t\\abs{\\nabla f(x)}\\cdot\\norm{\\nabla^2 f}_\\infty\n \\end{equation*}\n and this is the desired statement.\n\\end{proof}\n\n\\begin{remark}\n Let us emphasize that the only statement contained in \\cref{thm:hopflax_properties} that we are going to\n use is \\cref{it:hopflax_convexity}. Indeed it will be crucial when studying the stability of optimal\n maps. Furthermore, such a statement should be seen more like as a property of the $c$-conjugate \n (see \\cite[Section 1.2]{Santambrogio15}) than as a property of the Hopf-Lax semigroup.\n \n We have proven all other statements in order to give a complete reference on the short-time behavior of\n the Hopf-Lax semigroup when the initial datum is in $C^{1,1}(M)$.\n\\end{remark}\n\n\n\n\\section{Quantitative Stability of the Optimal Map}\\label{sec:stability}\nIn this section we will always refer to the optimal transport with respect to the quadratic\ncost between two probability measures in $\\prob(M)$ that are absolutely continuous with respect to the volume\nmeasure $\\m$ of a compact Riemannian manifold $(M, \\metric)$.\n\nThe duality theory of optimal transport can be seen as a tool to bound from above and from below \nthe optimal transport cost. Indeed, simply producing a transport map we can bound the cost from above, whereas\nwith a pair of potentials we can bound it from below. Estimating the optimal cost is the best one can \ndesire for a generic convex problem, but for the optimal transport problem we know that the optimal \nmap is unique (see \\cite{McCann01}) and thence we would like to be able to approximate it.\n\nIn details, we want to investigate the following problem.\n\\begin{problem}\n Let $\\nu, \\,\\mu_1,\\, \\mu_2\\in \\prob(M)$ be probability measures with $\\nu\\ll \\m$. Let $S,\\, T$ be \n the optimal transport maps from $\\nu$ to $\\mu_1$ and $\\mu_2$ respectively. \n Estimate the $L^2(\\nu)$-distance $\\norm{d(S, T)}^2_{L^2(\\nu)}$ between the two maps.\n\\end{problem}\nThe approach we are going to adopt builds upon the method, suggested to N.Gigli by the first author, who\nused it in \\cite[Proposition 3.3 and Corollary 3.4]{gigli2011}. \nIn the proof of the mentioned results, the author obtains (even if not stated in this way) exactly\nthe same inequality we are going to obtain. The substantial difference is that those results (and their proofs)\nwork only when the ambient is the Euclidean space. \n\nTransporting the proofs from the flat to the curved setting is not straight-forward. The proof of \nProposition~3.3 of the mentioned paper does not work on a Riemannian manifold, \nbecause curvature comes into play when comparing tangent vectors at different points.\nTo overcome this difficulty we have\ncome up with \\cref{it:hopflax_convexity} of \\cref{thm:hopflax_properties}. On the contrary, the proof of \nCorollary 3.4 is easily adapted on a compact Riemannian manifold.\n\nLet us also mention the recent result \\cite[Theorem 4.1]{berman2018}. In the said theorem the author\nobtain a quantitative stability of the optimal map when, instead of changing the target measure as we are\ndoing, the source measure is changed. The proof is totally different from ours and is mainly based on\ncomplex analytic tools. Also in that paper only the Euclidean setting (and the flat torus) is considered.\n\nWe will attack the stability problem only in the \\emph{perturbative setting}, namely when the optimal\nmap from $\\nu$ to $\\mu_1$ is the identity up to the first order.\nWorking only in the perturbative setting might look like an extremely strong assumption that would yield\nno applications at all. This is not the case, indeed what we call \\emph{perturbative setting} is \nmore or less equivalent to requiring only that the optimal transport map $T$ is local (meaning that \n$T-\\id$ is uniformly small) and well-behaved.\nFor example, and this is the whole point of \\cite{ambrosio-glaudo2018}, the optimal map from the reference \nmeasure to a random point cloud is (with high probability) a perturbation of the identity.\n\nWe don't need any hypothesis on the optimal map between $\\nu$ and $\\mu_2$.\n\n\\begin{theorem}\\label{thm:optimal_map_stability}\n Let $(M,\\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$) and let us \n denote by $\\m$ its volume measure.\n\n Let $\\nu, \\mu_1, \\mu_2\\in\\prob(M)$ be three probability measures with $\\nu\\ll\\m$ and let $S, T:M\\to M$ \n be the optimal transport maps respectively for the pairs of measures $(\\nu, \\mu_1)$ and $(\\nu, \\mu_2)$. \n We assume that $S=\\exp(\\nabla f)$ where $f:M\\to\\R$ is a \n $C^{1,1}$-function\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the null Neumann boundary\n conditions.} such that $\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty \\le c$ where $c=c(M)$ is \n the constant considered in the statement of \\cref{thm:hopflax_properties}.\n \n Then it holds\n \\begin{equation*}\n \\int_M d^2(S, T)\\de\\nu \\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) \\fullstop\n \\end{equation*}\n\\end{theorem}\n\\begin{proof}\n Let us consider a generic transport map $S':M\\to M$ from $\\nu$ to $\\mu_1$ and recall that,\n according to \\cite{Glaudo19}, if $c(M)$ is small enough, then the map $S$ is optimal.\n\n Given $x\\in M$, let us apply \\cref{it:hopflax_convexity} of \\cref{thm:hopflax_properties} with \n $y=S(x)$ and $y'=S'(x)$ and $t=1$\n \\begin{equation*}\n d^2(S(x),S'(x)) \n \\lesssim Q_1f(S(x))-Q_1f(S'(x)) + \\frac12\\left(d^2(x, S'(x))-d^2(x, S(x))\\right) \\fullstop\n \\end{equation*}\n Integrating this inequality with respect to $\\nu$ we obtain\n \\begin{equation}\\label{eq:stability_same_measures}\n \\int_M d^2(S, S')\\de\\nu \\lesssim \\norm{d(S', \\id)}_{L^2(\\nu)}^2 - W_2^2(\\nu, \\mu_1)\n \\end{equation}\n as the first two terms cancel thanks to the fact that both $S$ and $S'$ sends $\\nu$ into $\\mu_1$.\n \n We can now prove the main statement under the additional assumption that there exists an optimal map \n $R:M\\to M$ from $\\mu_2$ to $\\mu_1$. Applying \\cref{eq:stability_same_measures} with $S' = R\\circ T$ we get\n \\begin{equation}\\label{eq:temporary_ineq}\n \\int_M d^2(S, R\\circ T)\\de\\nu \\lesssim \\norm{d(R\\circ T, \\id)}_{L^2(\\nu)}^2 - W_2^2(\\nu, \\mu_1) \\fullstop\n \\end{equation}\n Thanks to the triangle inequality, it holds\n \\begin{align*}\n \\norm{d(R\\circ T, \\id)}_{L^2(\\nu)} \n &\\le \\norm{d(R\\circ T, T)}_{L^2(\\nu)} + \\norm{d(T, \\id)}_{L^2(\\nu)}\n = \\norm{d(R, \\id)}_{L^2(\\mu_2)} + W_2(\\nu, \\mu_2) \\\\\n &\\le 2 W_2(\\mu_1, \\mu_2) + W_2(\\nu, \\mu_1) \\fullstop\n \\end{align*}\n Applying this last inequality into \\cref{eq:temporary_ineq} yields\n \\begin{align*}\n \\int_M d^2(S, R\\circ T)\\de\\nu \n &\\lesssim \\left[2 W_2(\\mu_1, \\mu_2) + W_2(\\nu, \\mu_1)\\right]^2 - W_2^2(\\nu, \\mu_1) \\\\\n &\\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1)\n \\end{align*}\n and the desired statement follows from the triangle inequality\n \\begin{align*}\n \\int_M d^2(S, T) &\\lesssim \\int_M d^2(S, R\\circ T)\\de\\nu + \\int_M d^2(R\\circ T, T)\\de\\nu \\\\\n &\\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) + \\int_M d^2(R, \\id)\\de\\mu_2 \\\\\n &= 2 W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) \\fullstop\n \\end{align*}\n\n \n It remains to drop the assumption on the existence of the optimal map $R$. Given that our ambient manifold \n is compact, we can apply the nonquantitative strong stability (see \\cite[Corollary 5.23]{Villani08}). \n Let us take a sequence of absolutely continuous probability measures $\\mu_2^n$ that weakly converges \n to $\\mu_2$. \n Thanks to McCann's Theorem (see \\cite{McCann01}) the optimal map $R^n$ from $\\mu_2^n$ to $\\mu_1$ exists \n and thanks to the strong stability we know that the optimal maps $T^n$ from $\\nu$ to $\\mu_1^n$ converge \n strongly in $L^2(\\nu)$ to $T$. \n Hence it is readily seen that the result for $\\mu_2$ can be obtained by passing to the limit the \n result for $\\mu_2^n$. \n\\end{proof}\n\n\\begin{remark}\n The first part of the proof of \\cref{thm:optimal_map_stability} might seem a bit magical. Let us describe\n what is happening under the hood. \n \n The function $f$ is the Kantorovich potential of the couple $(\\nu, \\mu_1)$ and\n hence, by standard theory in optimal transport, it must be $c$-concave. \n \n Our hypotheses ensure us that it is not only $c$-concave, but even \\emph{strictly} $c$-concave. \n Furthermore, the theory we have developed on the Hopf-Lax semigroup tells us that even the other \n potential $f^c=Q_1 f$ is strictly $c$-concave (this is exactly \\cref{it:hopflax_convexity}).\n \n The result follows integrating the strict $c$-concavity inequality with respect to the measure\n $\\nu$.\n\\end{remark}\n\n\n\\begin{remark}\n The main use of \\cref{thm:optimal_map_stability} is the following one. \n Assume that the optimal map from $\\nu$ to $\\mu_1$ is local and well-behaved (this ensures the validity \n of the hypotheses of the theorem) and furthermore that $\\mu_2$ is much closer to $\\mu_1$ than to $\\nu$. \n In this situation, the theorem tells us\n \\begin{equation*}\n \\int_M d^2(S, T)\\de\\nu \\ll \\int_M d^2(S, \\id)\\de\\nu \\comma\n \\end{equation*}\n and this conveys exactly the information that $S$ approximates very well $T$. Notice also\n that the improvement from $C^{0,1\/2}$ dependence of \\cite{gigli2011} to the kind of Lipschitz dependence \n is due to the fact that we are working in a perturbative regime, close to the reference measure.\n\\end{remark}\n\n\\section{Optimal map in the random matching problem}\\label{sec:random_matching}\nWe want to apply our result on the stability of the optimal map in the perturbative setting to the \nsemi-discrete random matching problem. In this section we will work on a compact closed Riemannian manifold \n$(M, \\metric)$ of dimension $2$ (or the square $\\cc01^2$). \nWe will denote with $\\m$ the volume measure, with the implicit assumption that it is a probability.\n\nIn this setting, the semi-discrete random matching problem can be formulated as follows.\nFor a fixed $n\\in\\N$, consider $n$ independent random points $X_1, X_2, \\dots, X_n$ $\\m$-uniformly distributed\non $M$. Study the optimal transport map $T^n$ (with respect to the quadratic cost) from $\\m$ to the empirical \nmeasure $\\mu^n = \\frac1n\\sum_{i} \\delta_{X_i}$.\n\nSince we want to attack the problem applying \\cref{thm:optimal_map_stability}, first of all we have to choose \n$\\nu,\\, \\mu_1$ and $\\mu_2$.\nThe choices of $\\nu$ and $\\mu_2$ are very natural, indeed we set $\\nu=\\m$ and $\\mu_2=\\mu^n$. This way the map\n$T$ is $T^n$ .\n\nFar less obvious is the choice of $\\mu_1$, $S$ and $f$. As one might expect from the statement \nof \\cref{thm:main_theorem} and from the ansatz described in the introduction, our choice is $f=f^{n,t}$.\nThus $S=\\exp(\\nabla f^{n,t})$ (for some appropriate $t=t(n)$). \nFurthermore, keeping the same notation of \\cite{ambrosio-glaudo2018}, the measure $\\mu_1=S_\\#\\m$ will \nbe denoted by $\\hat\\mu^{n,t}$.\n\nFirst of all it is crucial to understand whether we are in position to apply \\cref{thm:optimal_map_stability}.\nIndeed we need to check if $\\nabla^2 f^{n,t}$ and $\\nabla f^{n,t}$ are sufficiently small. Moreover we have\nto obtain a strong estimate on $W_2^2(\\mu_1, \\mu_2)$. \nBoth this facts are among the main results obtained in \\cite{ambrosio-glaudo2018}. \nHence let us state them in the following proposition.\n\n\\begin{proposition}[Summary of results from \\texorpdfstring{\\cite{ambrosio-glaudo2018}}{[AG18]}]\\label{prop:ag18}\n Let $(M, \\metric)$ be a closed compact $2$-dimensional Riemannian manifold (or the square $\\cc01^2$) whose\n volume measure $\\m$ is a probability.\n Given $n\\in\\N$, let $X_1,\\dots, X_n$ be $n$ independent random points $\\m$-uniformly distributed on $M$\n and denote $\\mu^n=\\frac1n\\sum_i \\delta_{X_i}$ the associated empirical measure.\n \n For a choice of the time $t>0$, let $\\mu^{n,t}=P^*_t(\\mu^n)$ be the evolution through the heat flow of\n the empirical measure and let $f^{n,t}:M\\to\\R$ be the unique null-mean solution\\footnote{If $M=\\cc01^2$\n we ask also that $f$ satisfies the null Neumann boundary conditions.} to the Poisson equation\n $-\\lapl f^{n,t} = \\mu^{n,t}-1$.\n Finally, let us define the probability measure $\\hat\\mu^{n,t}$ as the push-forward of $\\m$ through the map\n $\\exp(\\nabla f^{n,t})$.\n \n For any $\\xi>0$, let $A^{n,t}_\\xi$ be the probabilistic event $\\{\\norm{\\nabla^2 f^{n,t}}_\\infty < \\xi\\}$.\n \n If $t=t(n)=\\frac{\\log^4(n)}{n}$ and $\\xi=\\xi(n) = \\frac1{\\log(n)}$, the following statements\\footnote{In \n \\cite{ambrosio-glaudo2018} the time $t(n)$ is chosen as $t(n)=\\gamma\\frac{\\log^3(n)}{n}$, where \n $\\gamma$ is a constant. \n As we clarify in \\cref{rem:time_is_flexible}, the choice of the exponent of the logarithm in the \n definition of $t(n)$ is not rigid. \n We choose the exponent $4$ instead of $3$ since it lets us get some estimates in a cleaner form and \n makes it possible to avoid inserting a constant in the definition of $t(n)$.}\n hold\n \\begin{itemize}\n \\item We know the asymptotic behavior of the expected matching cost\n \\begin{equation}\\label{eq:matching_cost_limit}\n \\lim_{n\\to\\infty} \\E{W_2^2(\\m, \\mu^n)}\\left(\\frac1{4\\pi}\\frac{\\log(n)}n\\right)^{-1} = 1 \\fullstop\n \\end{equation}\n \\item The probability of the complement of $A^{n,t}_\\xi$ decays faster than any power.\n In formulas, for any $k>0$ there exists a constant $C=C(M, k)$ such that \n \\begin{equation}\\label{eq:exceptional_set_rare}\n \\P{\\left(A^{n,t}_\\xi\\right)^\\complement} \n \\le C(M, k) n^{-k} \\fullstop\n \\end{equation}\n \\item One has the refined contractivity estimate\\footnote{This does not follow from the well-known \n contractivity property for the heat semigroup. Indeed the standard contractivity would yield \n an estimate of order $t=\\gamma\\frac{\\log^4(n)}{n}\\gg \\frac{\\log(n)}{n}$\n and such magnitude is too \n large for our purposes.}\n \\begin{equation}\\label{eq:diffusion_error}\n \\E{W_2^2(\\mu^n, \\mu^{n,t})} \\lesssim \\frac{\\log(\\log(n))}n \n \\left(\\ll \\frac{\\log(n)}{n}\\right)\\fullstop\n \\end{equation}\n \\item We are able to control the perturbation error with\n \\begin{equation}\\label{eq:perturbation_error}\n \\E{W_2^2(\\mu^{n,t}, \\hat\\mu^{n,t})} \\lesssim \\frac{1}{n\\log(n)} \n \\left(\\ll \\frac{\\log(n)}{n}\\right)\\fullstop\n \\end{equation}\n \\item \\label{it:exp_is_optimal}\n When $n$ is sufficiently large, in the event $A^{n,t}_\\xi$ the map $\\exp(\\nabla f^{n,t})$ is \n optimal from $\\m$ to $\\hat\\mu^{n,t}$.\n \\end{itemize}\n\\end{proposition}\n\\begin{proof}\n All of these results are contained in \\cite{ambrosio-glaudo2018} and thus we will only\n give a precise reference for them. All references are to propositions contained\n in \\cite{ambrosio-glaudo2018}.\n \n The validity of \\cref{eq:matching_cost_limit} is contained in Theorem 1.2.\n The fact that the event $A^{n,t}_\\xi$ has overwhelming probability follows from Theorem 3.3.\n The refined contractivity estimate \\cref{eq:diffusion_error} is Theorem 5.2.\n \n The estimate \\cref{eq:perturbation_error} follows from Equation 6.2 and Lemma 3.14. \n More specifically Equation 6.2 tells us that in the event $A^{n,t}_\\xi$ it holds\n \\begin{equation*}\n \\E{W_2^2(\\mu^{n,t}, \\hat\\mu^{n,t})} \\lesssim \\xi^2 \\int_M\\abs{\\nabla f^{n,t}}^2\\de\\m\n \\end{equation*}\n and Lemma 3.14 gives us the expected value of the Dirichlet energy of $f^{n,t}$. The behavior in the \n complementary of $A^{n,t}_\\xi$ can be ignored thanks to \\cref{eq:exceptional_set_rare}.\n \n It remains to show that in the event $A^{n,t}_\\xi$, the map $\\exp(\\nabla f^{n,t})$ is optimal. \n This follows directly from \\cite[Theorem 1.1]{Glaudo19}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:repeat_old_remark}\n Let us repeat the elementary observation made in \\cite[Remark 5.3]{ambrosio-glaudo2018}, as it will \n be useful.\n \n Let $X, Y$ be two random variables such that, in an event $E$, it holds $X\\le Y$. Then\n \\begin{equation*}\n \\E{X} \\le \\E{Y} + (\\norm{X}_\\infty + \\norm{Y}_\\infty)\\P{E^\\complement} \\fullstop\n \\end{equation*}\n In particular, if the infinity norm of $X, Y$ is suitably controlled and the probability of $E^\\complement$\n is exceedingly small, we can assume $\\E{X}\\le \\E{Y}$ up to a small error.\n \n This observation allows us to restrict our study to the \\emph{good} event $A^{n,t}_\\xi$. Indeed all \n quantities involved in our computations have at most polynomial growth, whereas \n $\\P{(A^{n,t}_\\xi)^\\complement}$ decays faster than any power.\n\\end{remark}\n\n\nOnce we have these results in our hands, the proof of the main theorem follows rather easily. Indeed\nwe just have to check that all assumptions of our stability result are satisfied.\n\n\\begin{proof}[Proof of \\cref{thm:main_theorem}]\n Let us assume to be in the event $A^{n,t}_\\xi$ with $\\xi = \\frac1{\\log(n)}$.\n Hence, thanks to \\cref{it:exp_is_optimal}, we can apply \\cref{thm:optimal_map_stability} to the triple\n of measures $\\nu=\\m$, $\\mu_1=\\hat\\mu^{n,t}$ and $\\mu_2=\\mu^n$ (with \n $S=\\exp(\\nabla f^{n,t})$ and $T=T^n$). \n We obtain\n \\begin{align*}\n \\int_M d^2(\\exp(\\nabla f^{n,t}), T^n)\\de\\m &\\lesssim \n W_2^2(\\mu^n, \\hat\\mu^{n,t}) + W_2(\\mu^n, \\hat\\mu^{n,t})W_2(\\m, \\hat\\mu^{n,t}) \\\\\n &\\lesssim \n W_2^2(\\mu^n, \\hat\\mu^{n,t}) + W_2(\\mu^n, \\hat\\mu^{n,t})W_2(\\m, \\mu^n) \n \\fullstop\n \\end{align*}\n \n Recalling \\cref{rem:repeat_old_remark} and \\cref{eq:exceptional_set_rare}, if we consider the expected \n value we can apply the latter inequality as if it were true unconditionally and not only in the \n event $A^{n,t}_\\xi$. Thus, taking the expected value and applying Cauchy-Schwarz's inequality, we get\n \\begin{equation*}\n \\E{\\int_M d^2(\\exp(\\nabla f^{n,t}), T^n)\\de\\m} \\lesssim \n \\E{W_2^2(\\mu^n, \\hat\\mu^{n,t})} \n + \\sqrt{\\E{W^2_2(\\mu^n, \\hat\\mu^{n,t})}\\cdot \\E{W^2_2(\\m, \\mu^n)}}\n \\fullstop\n \\end{equation*}\n The desired statement follows directly applying \n \\cref{eq:matching_cost_limit,eq:diffusion_error,eq:perturbation_error}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:time_is_flexible}\n It might seem that our choice of the time $t=\\log^4(n)\/n$ is a little arbitrary, and indeed it is. Any time\n $t=t(n)$ of order $\\log^\\alpha(n)\/n$, for some $\\alpha > 3$, would have worked flawlessly.\n\\end{remark}\n\nIt remains to justify \\cref{rem:distance-tangent}. As already said, the desired estimate boils down to\nthe validity of \n\\begin{equation}\\label{eq:linf_distance_bounded}\n \\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\ll \\frac{\\log(n)}n\n\\end{equation}\nfor any fixed $\\eps>0$.\nThe strategy of the proof is as follows. \nWith \\cref{lem:linf_l2_bound} (see also \\cite[Lemma 4.1]{goldman2017}) we reduce the hard task of \ncontrolling the $L^\\infty$-distance between $T^n$ and $\\id$ to the easier task of controlling \n$W_2^2(\\m,\\mu^n)$. \nThis latter estimate is then shown to be a consequence of \\cref{eq:exceptional_set_rare}.\n\n\\begin{lemma}\\label{lem:linf_l2_bound}\n Let $(M,\\metric)$ be a $d$-dimensional compact Riemannian manifold (possibly with Lipschitz boundary) and \n let $\\m$ be the volume measure on $M$.\n \n If $T:M\\to M$ is the optimal map with respect to the quadratic cost from $\\m$ to $T_\\#\\m$,\n then one has\n \\begin{equation*}\n \\norm{d(\\id, T)}_{L^\\infty(M)}\n \\lesssim \\left(\\int_M d^2(\\id, T)\\de\\m\\right)^{\\frac1{d+2}} \\fullstop\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Since the map $T$ is optimal, its graph is essentially contained in $c$-cyclically monotone set \n (see~\\cite[Theorem 1.38]{Santambrogio15}). \n More precisely, there exists a Borel set $C\\subseteq M$ such that $\\{(x,T(x)):\\ x\\in C\\}$ is $c$-cyclically \n monotone and $M\\setminus C$ is $\\m$-negligible. \n We will reduce our considerations to points in $C$ in order to exploit the $c$-cyclical monotonicity.\n\n Let us fix a point $x_0\\in C$ and let us define $\\alpha \\defeq \\frac12 d(x_0, T(x_0))$.\n Let us define the point $p\\in M$ as the middle point between $x_0$ and $T(x_0)$, that is \n $d(x_0, p) = d(p, T(x_0)) = \\alpha$. \n Let us consider a point $x\\in B(p, \\eps\\alpha)\\cap C$ where $\\eps>0$ is a small constant that will \n be chosen a posteriori. Finally let us define $\\beta\\defeq d(x, T(x))$. We want to show that \n $\\beta$ cannot be much smaller than $\\alpha$.\n \\begin{figure}[htb]\n \\centering\n \\input{linf_l2_bound.tikz}\n \\caption{The points considered in the proof of of \\cref{lem:linf_l2_bound}.}\n \\end{figure}\n \n \n Thanks to the $c$-cyclical monotonicity of $C$, it holds\n \\begin{equation*}\n d^2(x_0, T(x_0)) + d^2(x, T(x)) \\le d^2(x, T(x_0)) + d^2(x_0, T(x))\n \\end{equation*}\n and thus, applying repeatedly the triangle inequality, we deduce\n \\begin{align*}\n 4\\alpha^2 + \\beta^2 \n &\\le \\left(d(x, p) + d(p, T(x_0))\\right)^2 + \\left(d(x_0, p) + d(p, x) + d(x, T(x))\\right)^2 \\\\\n &\\le (\\eps\\alpha + \\alpha)^2 + (\\alpha + \\eps\\alpha + \\beta)^2 \n = 2(1+\\eps)^2\\alpha^2 + \\beta^2 + 2(1+\\eps)\\alpha\\beta \\\\\n &\\hphantom{asdasdasdasdasdasdasd}\\Updownarrow \\\\\n &\\hphantom{asdasdasd}(2-(1+\\eps)^2) \\alpha \\le (1+\\eps)\\beta \\fullstop\n \\end{align*}\n If $\\eps$ is chosen sufficiently small (i.e. $\\eps = 1\/3$), the desired estimate\n $\\alpha\\lesssim\\beta$ follows.\n \n Since $x$ can be chosen arbitrarily in $B(p, \\eps\\alpha)\\cap C$, the estimate \n $\\alpha\\lesssim \\beta$ implies\n \\begin{align*}\n \\int_M d^2(x, T(x))\\de\\m(x) \n &\\ge \\int_{B(p, \\eps\\alpha)} d^2(x, T(x))\\de\\m(x)\n \\gtrsim \\m(B(p, \\eps\\alpha)) d^2(x_0, T(x_0)) \\\\\n &\\gtrsim \\eps\\alpha^d d^2(x_0, T(x_0)\n \\gtrsim \\bigl(d(x_0, T(x_0))\\bigr)^{d+2} \\comma\n \\end{align*}\n where we have used that a ball with radius $r$ not larger than the diameter of \n $M$ has measure comparable to $r^d$ (follows from the Ahlfors-regularity of compact \n Riemannian manifolds with Lipschitz boundary).\n This completes the proof since $x_0$ can be chosen arbitrarily in a set with full measure.\n\\end{proof}\n\\begin{remark}\n The previous lemma holds, with the same proof, on any Ahlfors-regular metric measure space \n that is also a length space.\n\\end{remark}\n\\begin{remark}\n If we apply \\cref{lem:linf_l2_bound} on a $2$-dimensional manifold with $T^n$ being the optimal map\n (with respect to the quadratic cost) from $\\m$ to the empirical measure $\\mu^n$, we obtain\n \\begin{equation*}\n \\norm{d(\\id,T^n}_{L^{\\infty}(M)} \n \\lesssim W_2(\\m, \\mu^n)^{\\frac{1}{2}} \\fullstop\n \\end{equation*}\n Since we know (as a consequence of \\cref{eq:limit_value}) that with high probability\n $W^2_2(\\m, \\mu^n) \\lesssim n^{-1}\\log(n)$, we deduce that with high probability\n it holds\n \\begin{equation*}\n \\norm{d(\\id,T^n}_{L^{\\infty}(M)} \\lesssim \\left(\\frac{\\log(n)}{n}\\right)^{\\frac14} \\fullstop\n \\end{equation*}\n This estimate does not match the asymptotic behavior of the $\\infty$-Wasserstein distance between $\\m$\n and $\\mu^n$. In fact, as proven in \\cite{leighton1989,shor1991,trillos2015}, with high probability \n it holds\n \\begin{equation*}\n W_{\\infty}(\\m, \\mu^n) \\approx \\frac{\\log(n)^{\\frac34}}{n^{\\frac12}} \\fullstop\n \\end{equation*}\n\\end{remark}\n\nWe are now ready to show \\cref{eq:linf_distance_bounded} (to be precise we prove a much stronger \nestimate).\n\\begin{proposition}\\label{prop:linf_is_small}\n Using the same notation and definitions of the statement of \\cref{thm:main_theorem}, for any \n $\\eps>0$ and any $k>0$ there exists a constant $C=C(M, \\eps, k)$ such that\n \\begin{equation}\n \\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\le C(M, \\eps, k)n^{-k} \\fullstop\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\n We show that for any $\\eps>0$ and any $k>0$ there exists a constant $C=C(M, \\eps, k)$\n such that\n \\begin{equation}\\label{eq:wasserstein_is_small}\n \\P{W_2(\\m, \\mu^n) > \\varepsilon} \\le C(M, \\eps, k)n^{-k} \\fullstop\n \\end{equation}\n In fact, if we are able to prove \\cref{eq:wasserstein_is_small}, then the statement of the proposition \n follows applying \\cref{lem:linf_l2_bound} with $T=T^n$ (changing adequately $\\eps,k$ and the value\n of the constant $C$).\n \n The triangle inequality gives us\n \\begin{equation}\\label{eq:linf_tmp1}\n W_2(\\m, \\mu^n) \\le W_2(\\mu^{n,t}, \\mu^n) + W_2(\\m, \\mu^{n,t}) \\fullstop\n \\end{equation}\n The first term can be bounded using the contractivity property of the heat semigroup, obtaining\n \\begin{equation}\\label{eq:linf_tmp2}\n W_2(\\mu^{n,t}, \\mu^n) \\lesssim \\sqrt{t} \\fullstop\n \\end{equation}\n For the second term we employ the transport inequality \\cite[(4.1)]{ambrosio-glaudo2018} and get\n \\begin{equation}\\label{eq:linf_tmp3}\n W^2_2(\\m, \\mu^{n,t}) \\lesssim \\int_M \\abs{\\nabla f^{n,t}}^2\\de\\m \\fullstop\n \\end{equation}\n If we assume to be in the event $A^{n,t}_\\xi$ (that is defined in the statement of \\cref{prop:ag18})\n with $\\xi=\\xi(n)=\\frac1{\\log(n)}$, we have\n \\begin{equation}\\label{eq:linf_tmp4}\n \\int_M \\abs{\\nabla f^{n,t}}^2\\de\\m \\lesssim \\norm{\\nabla f}_{L^{\\infty}(M)}^2\n \\lesssim \\norm{\\nabla^2 f}_{L^{\\infty}(M)}^2 \\le \\xi^2 \\fullstop\n \\end{equation}\n Joining \\cref{eq:linf_tmp1,eq:linf_tmp2,eq:linf_tmp3,eq:linf_tmp4} we deduce that in the event \n $A^{n,t}_\\xi$ it holds\n \\begin{equation*}\n W_2(\\m, \\mu^n) \\lesssim \\sqrt{t} + \\xi \\fullstop\n \\end{equation*}\n Since $t(n)\\to 0$ and $\\xi(n)\\to 0$ as $n\\to\\infty$, this implies (for $n$ sufficiently large) that\n in the event $A^{n,t}_\\xi$ it holds $W_2(\\m, \\mu^n) \\le \\eps$. Hence \\cref{eq:wasserstein_is_small}\n is a consequence of \\cref{eq:exceptional_set_rare} and this concludes of the proof.\n\\end{proof}\n\n\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}