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|---|---|---|---|---|---|---|
R : Type u
inst✝⁸ : CommSemiring R
A : Type u
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
B : Type u
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FormallyUnramified R A
inst✝ : FormallyUnramified A B
⊢ FormallyUnramified R B
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
|
constructor
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁸ : CommSemiring R
A : Type u
inst✝⁷ : CommSemiring A
inst✝⁶ : Algebra R A
B : Type u
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R B
inst✝³ : Algebra A B
inst✝² : IsScalarTower R A B
inst✝¹ : FormallyUnramified R A
inst✝ : FormallyUnramified A B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra R B_1] (I : Ideal B_1),
I ^ 2 = ⊥ → Function.Injective (AlgHom.comp (Ideal.Quotient.mkₐ R I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
|
intro C _ _ I hI f₁ f₂ e
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
|
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
⊢ AlgHom.comp (Ideal.Quotient.mkₐ R I) (AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B)) =
AlgHom.comp (Ideal.Quotient.mkₐ R I) (AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by
|
rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc]
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
|
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
|
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
F₁ : B →ₐ[A] C :=
{ toRingHom := ↑f₁,
commutes' :=
(_ : ∀ (r : A), OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r) = OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r)) }
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
|
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
F₁ : B →ₐ[A] C :=
{ toRingHom := ↑f₁,
commutes' :=
(_ : ∀ (r : A), OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r) = OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r)) }
F₂ : B →ₐ[A] C :=
{ toRingHom := ↑f₂,
commutes' :=
(_ :
∀ (x : A),
(AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)) x = (AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B)) x) }
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
|
ext1 x
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective.H
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
F₁ : B →ₐ[A] C :=
{ toRingHom := ↑f₁,
commutes' :=
(_ : ∀ (r : A), OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r) = OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r)) }
F₂ : B →ₐ[A] C :=
{ toRingHom := ↑f₂,
commutes' :=
(_ :
∀ (x : A),
(AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)) x = (AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B)) x) }
x : B
⊢ f₁ x = f₂ x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
|
change F₁ x = F₂ x
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective.H
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
F₁ : B →ₐ[A] C :=
{ toRingHom := ↑f₁,
commutes' :=
(_ : ∀ (r : A), OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r) = OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r)) }
F₂ : B →ₐ[A] C :=
{ toRingHom := ↑f₂,
commutes' :=
(_ :
∀ (x : A),
(AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)) x = (AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B)) x) }
x : B
⊢ F₁ x = F₂ x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
|
congr
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
case comp_injective.H.e_a
R : Type u
inst✝¹⁰ : CommSemiring R
A : Type u
inst✝⁹ : CommSemiring A
inst✝⁸ : Algebra R A
B : Type u
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R B
inst✝⁵ : Algebra A B
inst✝⁴ : IsScalarTower R A B
inst✝³ : FormallyUnramified R A
inst✝² : FormallyUnramified A B
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[R] C
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
e' : AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B) = AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)
this : Algebra A C := RingHom.toAlgebra ↑(AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B))
F₁ : B →ₐ[A] C :=
{ toRingHom := ↑f₁,
commutes' :=
(_ : ∀ (r : A), OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r) = OneHom.toFun (↑↑↑f₁) ((algebraMap A B) r)) }
F₂ : B →ₐ[A] C :=
{ toRingHom := ↑f₂,
commutes' :=
(_ :
∀ (x : A),
(AlgHom.comp f₂ (IsScalarTower.toAlgHom R A B)) x = (AlgHom.comp f₁ (IsScalarTower.toAlgHom R A B)) x) }
x : B
⊢ F₁ = F₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
|
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
|
Mathlib.RingTheory.Etale.288_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B
|
Mathlib_RingTheory_Etale
|
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : CommSemiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : Algebra A B
inst✝¹ : IsScalarTower R A B
inst✝ : FormallyUnramified R B
⊢ FormallyUnramified A B
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
|
constructor
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : CommSemiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : Algebra A B
inst✝¹ : IsScalarTower R A B
inst✝ : FormallyUnramified R B
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra A B_1] (I : Ideal B_1),
I ^ 2 = ⊥ → Function.Injective (AlgHom.comp (Ideal.Quotient.mkₐ A I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
|
intro Q _ _ I e f₁ f₂ e'
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
|
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
this : Algebra R Q := RingHom.toAlgebra (RingHom.comp (algebraMap A Q) (algebraMap R A))
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
|
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
this✝ : Algebra R Q := RingHom.toAlgebra (RingHom.comp (algebraMap A Q) (algebraMap R A))
this : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
|
refine' AlgHom.restrictScalars_injective R _
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
this✝ : Algebra R Q := RingHom.toAlgebra (RingHom.comp (algebraMap A Q) (algebraMap R A))
this : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
⊢ AlgHom.restrictScalars R f₁ = AlgHom.restrictScalars R f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
|
refine' FormallyUnramified.ext I ⟨2, e⟩ _
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
this✝ : Algebra R Q := RingHom.toAlgebra (RingHom.comp (algebraMap A Q) (algebraMap R A))
this : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
⊢ ∀ (x : B),
(Ideal.Quotient.mk I) ((AlgHom.restrictScalars R f₁) x) = (Ideal.Quotient.mk I) ((AlgHom.restrictScalars R f₂) x)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
|
intro x
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁹ : CommSemiring R
A : Type u
inst✝⁸ : CommSemiring A
inst✝⁷ : Algebra R A
B : Type u
inst✝⁶ : Semiring B
inst✝⁵ : Algebra R B
inst✝⁴ : Algebra A B
inst✝³ : IsScalarTower R A B
inst✝² : FormallyUnramified R B
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra A Q
I : Ideal Q
e : I ^ 2 = ⊥
f₁ f₂ : B →ₐ[A] Q
e' : AlgHom.comp (Ideal.Quotient.mkₐ A I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ A I) f₂
this✝ : Algebra R Q := RingHom.toAlgebra (RingHom.comp (algebraMap A Q) (algebraMap R A))
this : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
x : B
⊢ (Ideal.Quotient.mk I) ((AlgHom.restrictScalars R f₁) x) = (Ideal.Quotient.mk I) ((AlgHom.restrictScalars R f₂) x)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
|
exact AlgHom.congr_fun e' x
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
|
Mathlib.RingTheory.Etale.304_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
⊢ FormallySmooth R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
|
constructor
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
⊢ ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
I ^ 2 = ⊥ → Function.Surjective (AlgHom.comp (Ideal.Quotient.mkₐ R I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
|
intro C _ _ I hI i
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = i
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
|
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
⊢ P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
|
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
⊢ ∀ a ∈ RingHom.ker ↑f ^ 2, (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) a = 0
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
|
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
⊢ RingHom.ker f ≤ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
|
rintro x (hx : f x = 0)
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
x : P
hx : f x = 0
⊢ x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
|
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
x : P
hx : f x = 0
this : (Ideal.Quotient.mk I) ((lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) x) = i (f x)
⊢ x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
|
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
this : RingHom.ker f ≤ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
⊢ ∀ a ∈ RingHom.ker ↑f ^ 2, (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) a = 0
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
|
intro x hx
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
this : RingHom.ker f ≤ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
x : P
hx : x ∈ RingHom.ker ↑f ^ 2
⊢ (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) x = 0
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
|
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
this✝ : RingHom.ker f ≤ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) I
x : P
hx : x ∈ RingHom.ker ↑f ^ 2
this : x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) (I ^ 2)
⊢ (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) x = 0
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
|
rwa [hI] at this
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = i
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
|
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
⊢ AlgHom.comp i (AlgHom.kerSquareLift f) = AlgHom.comp (Ideal.Quotient.mkₐ R I) l
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
|
apply AlgHom.coe_ringHom_injective
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case a
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
⊢ ↑(AlgHom.comp i (AlgHom.kerSquareLift f)) = ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) l)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
|
apply Ideal.Quotient.ringHom_ext
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case a.h
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
⊢ RingHom.comp (↑(AlgHom.comp i (AlgHom.kerSquareLift f))) (Ideal.Quotient.mk (RingHom.ker ↑f ^ 2)) =
RingHom.comp (↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) l)) (Ideal.Quotient.mk (RingHom.ker ↑f ^ 2))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
|
ext x
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case a.h.a
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
x : P
⊢ (RingHom.comp (↑(AlgHom.comp i (AlgHom.kerSquareLift f))) (Ideal.Quotient.mk (RingHom.ker ↑f ^ 2))) x =
(RingHom.comp (↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) l)) (Ideal.Quotient.mk (RingHom.ker ↑f ^ 2))) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
|
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
this : AlgHom.comp i (AlgHom.kerSquareLift f) = AlgHom.comp (Ideal.Quotient.mkₐ R I) l
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = i
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
|
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁸ : CommRing R
inst✝⁷ : CommSemiring S
P A : Type u
inst✝⁶ : CommRing A
inst✝⁵ : Algebra R A
inst✝⁴ : CommRing P
inst✝³ : Algebra R P
I✝ : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝² : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra R C
I : Ideal C
hI : I ^ 2 = ⊥
i : A →ₐ[R] C ⧸ I
l : P ⧸ RingHom.ker ↑f ^ 2 →ₐ[R] C :=
Ideal.Quotient.liftₐ (RingHom.ker ↑f ^ 2) (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f))
(_ : ∀ x ∈ RingHom.ker ↑f ^ 2, x ∈ Ideal.comap (lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp i f)) ⊥)
this : AlgHom.comp i (AlgHom.kerSquareLift f) = AlgHom.comp (Ideal.Quotient.mkₐ R I) l
⊢ AlgHom.comp (Ideal.Quotient.mkₐ R I) (AlgHom.comp l g) = i
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by
|
rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by
|
Mathlib.RingTheory.Etale.330_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
⊢ FormallySmooth R A ↔ ∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
|
constructor
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
⊢ FormallySmooth R A → ∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
·
|
intro
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
·
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
⊢ ∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
|
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
⊢ ∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
|
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
⊢ RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
|
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
⊢ ∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
|
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
⊢ AlgHom.comp (AlgHom.kerSquareLift f)
(lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj))) =
AlgHom.id R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
|
ext x
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp.H
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
⊢ (AlgHom.comp (AlgHom.kerSquareLift f)
(lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj))))
x =
(AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
|
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp.H
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
this :
↑(Ideal.quotientKerAlgEquivOfSurjective surj)
((Ideal.Quotient.mk (RingHom.ker ↑(AlgHom.kerSquareLift f)))
((lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)))
x)) =
↑(Ideal.quotientKerAlgEquivOfSurjective surj) (↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)) x)
⊢ (AlgHom.comp (AlgHom.kerSquareLift f)
(lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj))))
x =
(AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
|
erw [AlgEquiv.apply_symm_apply] at this
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mp.H
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
this :
↑(Ideal.quotientKerAlgEquivOfSurjective surj)
((Ideal.Quotient.mk (RingHom.ker ↑(AlgHom.kerSquareLift f)))
((lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)))
x)) =
x
⊢ (AlgHom.comp (AlgHom.kerSquareLift f)
(lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj))))
x =
(AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
|
conv_rhs => rw [← this, AlgHom.id_apply]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
this :
↑(Ideal.quotientKerAlgEquivOfSurjective surj)
((Ideal.Quotient.mk (RingHom.ker ↑(AlgHom.kerSquareLift f)))
((lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)))
x)) =
x
| (AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
rw [← this, AlgHom.id_apply]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
this :
↑(Ideal.quotientKerAlgEquivOfSurjective surj)
((Ideal.Quotient.mk (RingHom.ker ↑(AlgHom.kerSquareLift f)))
((lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)))
x)) =
x
| (AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
rw [← this, AlgHom.id_apply]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
a✝ : FormallySmooth R A
surj : Function.Surjective ⇑(AlgHom.kerSquareLift f)
sqz : RingHom.ker ↑(AlgHom.kerSquareLift f) ^ 2 = 0
x : A
this :
↑(Ideal.quotientKerAlgEquivOfSurjective surj)
((Ideal.Quotient.mk (RingHom.ker ↑(AlgHom.kerSquareLift f)))
((lift (RingHom.ker ↑(AlgHom.kerSquareLift f)) (_ : ∃ n, RingHom.ker ↑(AlgHom.kerSquareLift f) ^ n = 0)
↑(AlgEquiv.symm (Ideal.quotientKerAlgEquivOfSurjective surj)))
x)) =
x
| (AlgHom.id R A) x
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
rw [← this, AlgHom.id_apply]
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs =>
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mpr
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
⊢ (∃ g, AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A) → FormallySmooth R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
·
|
rintro ⟨g, hg⟩
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
·
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
case mpr.intro
R S : Type u
inst✝⁶ : CommRing R
inst✝⁵ : CommSemiring S
P A : Type u
inst✝⁴ : CommRing A
inst✝³ : Algebra R A
inst✝² : CommRing P
inst✝¹ : Algebra R P
I : Ideal P
f : P →ₐ[R] A
hf : Function.Surjective ⇑f
inst✝ : FormallySmooth R P
g : A →ₐ[R] P ⧸ RingHom.ker ↑f ^ 2
hg : AlgHom.comp (AlgHom.kerSquareLift f) g = AlgHom.id R A
⊢ FormallySmooth R A
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩;
|
exact FormallySmooth.of_split f g hg
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩;
|
Mathlib.RingTheory.Etale.351_0.sEffwLG8zJBnQIt
|
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
⊢ Subsingleton (Ω[S⁄R])
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
|
rw [← not_nontrivial_iff_subsingleton]
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
⊢ ¬Nontrivial (Ω[S⁄R])
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
|
intro h
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
⊢ False
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
|
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mk.intro.intro
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
f₁ f₂ : { f // AlgHom.comp (AlgHom.kerSquareLift (TensorProduct.lmul' R)) f = AlgHom.id R S }
e : f₁ ≠ f₂
⊢ False
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
|
apply e
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mk.intro.intro
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
f₁ f₂ : { f // AlgHom.comp (AlgHom.kerSquareLift (TensorProduct.lmul' R)) f = AlgHom.id R S }
e : f₁ ≠ f₂
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
|
ext1
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mk.intro.intro.a
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
f₁ f₂ : { f // AlgHom.comp (AlgHom.kerSquareLift (TensorProduct.lmul' R)) f = AlgHom.id R S }
e : f₁ ≠ f₂
⊢ ↑f₁ = ↑f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
|
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
f₁ f₂ : { f // AlgHom.comp (AlgHom.kerSquareLift (TensorProduct.lmul' R)) f = AlgHom.id R S }
e : f₁ ≠ f₂
⊢ IsNilpotent (RingHom.ker ↑(AlgHom.kerSquareLift (TensorProduct.lmul' R)))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
|
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : FormallyUnramified R S
h : Nontrivial (Ω[S⁄R])
f₁ f₂ : { f // AlgHom.comp (AlgHom.kerSquareLift (TensorProduct.lmul' R)) f = AlgHom.id R S }
e : f₁ ≠ f₂
⊢ IsNilpotent (Ideal.cotangentIdeal (RingHom.ker ↑(TensorProduct.lmul' R)))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
|
exact ⟨_, Ideal.cotangentIdeal_square _⟩
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
|
Mathlib.RingTheory.Etale.397_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
|
constructor
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ FormallyUnramified R S → Subsingleton (Ω[S⁄R])
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
·
|
intros
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
·
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mp
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
a✝ : FormallyUnramified R S
⊢ Subsingleton (Ω[S⁄R])
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros;
|
infer_instance
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros;
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Subsingleton (Ω[S⁄R]) → FormallyUnramified R S
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
·
|
intro H
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
·
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
H : Subsingleton (Ω[S⁄R])
⊢ FormallyUnramified R S
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
|
constructor
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr.comp_injective
R S : Type u
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
H : Subsingleton (Ω[S⁄R])
⊢ ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
I ^ 2 = ⊥ → Function.Injective (AlgHom.comp (Ideal.Quotient.mkₐ R I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
|
intro B _ _ I hI f₁ f₂ e
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr.comp_injective
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
H : Subsingleton (Ω[S⁄R])
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f₁ f₂ : S →ₐ[R] B
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
|
letI := f₁.toRingHom.toAlgebra
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr.comp_injective
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
H : Subsingleton (Ω[S⁄R])
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f₁ f₂ : S →ₐ[R] B
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
this : Algebra S B := RingHom.toAlgebra ↑f₁
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
|
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr.comp_injective
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
H : Subsingleton (Ω[S⁄R])
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f₁ f₂ : S →ₐ[R] B
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
this✝ : Algebra S B := RingHom.toAlgebra ↑f₁
this : IsScalarTower R S B
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
|
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
case mpr.comp_injective
R S : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
H : Subsingleton (Ω[S⁄R])
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra R B
I : Ideal B
hI : I ^ 2 = ⊥
f₁ f₂ : S →ₐ[R] B
e : AlgHom.comp (Ideal.Quotient.mkₐ R I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ R I) f₂
this✝¹ : Algebra S B := RingHom.toAlgebra ↑f₁
this✝ : IsScalarTower R S B
this : Subsingleton { f // AlgHom.comp (Ideal.Quotient.mkₐ R I) f = IsScalarTower.toAlgHom R S (B ⧸ I) }
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
|
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
|
Mathlib.RingTheory.Etale.409_0.sEffwLG8zJBnQIt
|
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R])
|
Mathlib_RingTheory_Etale
|
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u
inst✝² : CommSemiring B
inst✝¹ : Algebra R B
inst✝ : FormallyUnramified R A
⊢ FormallyUnramified B (B ⊗[R] A)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
|
constructor
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u
inst✝² : CommSemiring B
inst✝¹ : Algebra R B
inst✝ : FormallyUnramified R A
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra B B_1] (I : Ideal B_1),
I ^ 2 = ⊥ → Function.Injective (AlgHom.comp (Ideal.Quotient.mkₐ B I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
|
intro C _ _ I hI f₁ f₂ e
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B ⊗[R] A →ₐ[B] C
e : AlgHom.comp (Ideal.Quotient.mkₐ B I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ B I) f₂
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
|
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B ⊗[R] A →ₐ[B] C
e : AlgHom.comp (Ideal.Quotient.mkₐ B I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ B I) f₂
this : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B ⊗[R] A →ₐ[B] C
e : AlgHom.comp (Ideal.Quotient.mkₐ B I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ B I) f₂
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ f₁ = f₂
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
ext : 1
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective.ha
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B ⊗[R] A →ₐ[B] C
e : AlgHom.comp (Ideal.Quotient.mkₐ B I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ B I) f₂
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ AlgHom.comp f₁ TensorProduct.includeLeft = AlgHom.comp f₂ TensorProduct.includeLeft
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
·
|
exact Subsingleton.elim _ _
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
·
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_injective.hb
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallyUnramified R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f₁ f₂ : B ⊗[R] A →ₐ[B] C
e : AlgHom.comp (Ideal.Quotient.mkₐ B I) f₁ = AlgHom.comp (Ideal.Quotient.mkₐ B I) f₂
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ AlgHom.comp (AlgHom.restrictScalars R f₁) TensorProduct.includeRight =
AlgHom.comp (AlgHom.restrictScalars R f₂) TensorProduct.includeRight
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
·
|
exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
·
|
Mathlib.RingTheory.Etale.436_0.sEffwLG8zJBnQIt
|
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u
inst✝² : CommSemiring B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
⊢ FormallySmooth B (B ⊗[R] A)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
|
constructor
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R : Type u
inst✝⁵ : CommSemiring R
A : Type u
inst✝⁴ : Semiring A
inst✝³ : Algebra R A
B : Type u
inst✝² : CommSemiring B
inst✝¹ : Algebra R B
inst✝ : FormallySmooth R A
⊢ ∀ ⦃B_1 : Type u⦄ [inst : CommRing B_1] [inst_1 : Algebra B B_1] (I : Ideal B_1),
I ^ 2 = ⊥ → Function.Surjective (AlgHom.comp (Ideal.Quotient.mkₐ B I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
|
intro C _ _ I hI f
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ B I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
|
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ B I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ B I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_1
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ A →ₐ[R] C
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
·
|
exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
·
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_2
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ AlgHom.comp (Ideal.Quotient.mkₐ B I)
(TensorProduct.productLeftAlgHom (ofId B C)
(lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp (AlgHom.restrictScalars R f) TensorProduct.includeRight))) =
f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
·
|
apply AlgHom.restrictScalars_injective R
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
·
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_2.a
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ AlgHom.restrictScalars R
(AlgHom.comp (Ideal.Quotient.mkₐ B I)
(TensorProduct.productLeftAlgHom (ofId B C)
(lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp (AlgHom.restrictScalars R f) TensorProduct.includeRight)))) =
AlgHom.restrictScalars R f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
|
apply TensorProduct.ext'
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_2.a.H
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
⊢ ∀ (a : B) (b : A),
(AlgHom.restrictScalars R
(AlgHom.comp (Ideal.Quotient.mkₐ B I)
(TensorProduct.productLeftAlgHom (ofId B C)
(lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp (AlgHom.restrictScalars R f) TensorProduct.includeRight)))))
(a ⊗ₜ[R] b) =
(AlgHom.restrictScalars R f) (a ⊗ₜ[R] b)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
|
intro b a
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_2.a.H
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
b : B
a : A
⊢ (AlgHom.restrictScalars R
(AlgHom.comp (Ideal.Quotient.mkₐ B I)
(TensorProduct.productLeftAlgHom (ofId B C)
(lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp (AlgHom.restrictScalars R f) TensorProduct.includeRight)))))
(b ⊗ₜ[R] a) =
(AlgHom.restrictScalars R f) (b ⊗ₜ[R] a)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
|
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝¹ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this✝ : IsScalarTower R B C
b : B
a : A
this : (algebraMap B (C ⧸ I)) b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a)
⊢ (AlgHom.restrictScalars R
(AlgHom.comp (Ideal.Quotient.mkₐ B I)
(TensorProduct.productLeftAlgHom (ofId B C)
(lift I (_ : ∃ n, I ^ n = 0) (AlgHom.comp (AlgHom.restrictScalars R f) TensorProduct.includeRight)))))
(b ⊗ₜ[R] a) =
(AlgHom.restrictScalars R f) (b ⊗ₜ[R] a)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by
|
simpa [Algebra.ofId_apply]
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
case comp_surjective.refine'_2.a.H
R : Type u
inst✝⁷ : CommSemiring R
A : Type u
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
B : Type u
inst✝⁴ : CommSemiring B
inst✝³ : Algebra R B
inst✝² : FormallySmooth R A
C : Type u
inst✝¹ : CommRing C
inst✝ : Algebra B C
I : Ideal C
hI : I ^ 2 = ⊥
f : B ⊗[R] A →ₐ[B] C ⧸ I
this✝ : Algebra R C := RingHom.toAlgebra (RingHom.comp (algebraMap B C) (algebraMap R B))
this : IsScalarTower R B C
b : B
a : A
⊢ (algebraMap B (C ⧸ I)) b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
|
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
|
Mathlib.RingTheory.Etale.447_0.sEffwLG8zJBnQIt
|
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A)
|
Mathlib_RingTheory_Etale
|
R S Rₘ Sₘ : Type u
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing Rₘ
inst✝⁹ : CommRing Sₘ
M : Submonoid R
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R Sₘ
inst✝⁶ : Algebra S Sₘ
inst✝⁵ : Algebra R Rₘ
inst✝⁴ : Algebra Rₘ Sₘ
inst✝³ : IsScalarTower R Rₘ Sₘ
inst✝² : IsScalarTower R S Sₘ
inst✝¹ : IsLocalization M Rₘ
inst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
⊢ FormallySmooth R Rₘ
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
|
constructor
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S Rₘ Sₘ : Type u
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing Rₘ
inst✝⁹ : CommRing Sₘ
M : Submonoid R
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R Sₘ
inst✝⁶ : Algebra S Sₘ
inst✝⁵ : Algebra R Rₘ
inst✝⁴ : Algebra Rₘ Sₘ
inst✝³ : IsScalarTower R Rₘ Sₘ
inst✝² : IsScalarTower R S Sₘ
inst✝¹ : IsLocalization M Rₘ
inst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
⊢ ∀ ⦃B : Type u⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),
I ^ 2 = ⊥ → Function.Surjective (AlgHom.comp (Ideal.Quotient.mkₐ R I))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
|
intro Q _ _ I e f
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
|
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
⊢ ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
|
intro x
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
x : ↥M
⊢ IsUnit ((algebraMap R Q) ↑x)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
|
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
x : ↥M
⊢ IsUnit ((Ideal.Quotient.mk I) ((algebraMap R Q) ↑x))
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
|
convert (IsLocalization.map_units Rₘ x).map f
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case h.e'_3
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
x : ↥M
⊢ (Ideal.Quotient.mk I) ((algebraMap R Q) ↑x) = f ((algebraMap R Rₘ) ↑x)
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
|
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
|
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case comp_surjective
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
this : Rₘ →ₐ[R] Q :=
let src := IsLocalization.lift this✝;
{
toRingHom :=
{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
map_add' :=
(_ : ∀ (x y : Rₘ), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
commutes' := (_ : ∀ (x : R), (IsLocalization.lift this✝) ((algebraMap R Rₘ) x) = (algebraMap R Q) x) }
⊢ ∃ a, AlgHom.comp (Ideal.Quotient.mkₐ R I) a = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
|
use this
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case h
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
this : Rₘ →ₐ[R] Q :=
let src := IsLocalization.lift this✝;
{
toRingHom :=
{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
map_add' :=
(_ : ∀ (x y : Rₘ), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
commutes' := (_ : ∀ (x : R), (IsLocalization.lift this✝) ((algebraMap R Rₘ) x) = (algebraMap R Q) x) }
⊢ AlgHom.comp (Ideal.Quotient.mkₐ R I) this = f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
use this
|
apply AlgHom.coe_ringHom_injective
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
use this
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
case h.a
R S Rₘ Sₘ : Type u
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : CommRing Rₘ
inst✝¹¹ : CommRing Sₘ
M : Submonoid R
inst✝¹⁰ : Algebra R S
inst✝⁹ : Algebra R Sₘ
inst✝⁸ : Algebra S Sₘ
inst✝⁷ : Algebra R Rₘ
inst✝⁶ : Algebra Rₘ Sₘ
inst✝⁵ : IsScalarTower R Rₘ Sₘ
inst✝⁴ : IsScalarTower R S Sₘ
inst✝³ : IsLocalization M Rₘ
inst✝² : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ
Q : Type u
inst✝¹ : CommRing Q
inst✝ : Algebra R Q
I : Ideal Q
e : I ^ 2 = ⊥
f : Rₘ →ₐ[R] Q ⧸ I
this✝ : ∀ (x : ↥M), IsUnit ((algebraMap R Q) ↑x)
this : Rₘ →ₐ[R] Q :=
let src := IsLocalization.lift this✝;
{
toRingHom :=
{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
map_add' :=
(_ : ∀ (x y : Rₘ), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
commutes' := (_ : ∀ (x : R), (IsLocalization.lift this✝) ((algebraMap R Rₘ) x) = (algebraMap R Q) x) }
⊢ ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) this) = ↑f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Kaehler
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
/-!
# Formally étale morphisms
An `R`-algebra `A` is formally étale (resp. unramified, smooth) if for every `R`-algebra,
every square-zero ideal `I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists
exactly (resp. at most, at least) one lift `A →ₐ[R] B`.
We show that the property extends onto nilpotent ideals, and that these properties are stable
under `R`-algebra homomorphisms and compositions.
-/
-- Porting note: added to make the syntax work below.
open scoped TensorProduct
universe u
namespace Algebra
section
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [Semiring A] [Algebra R A]
variable {B : Type u} [CommRing B] [Algebra R B] (I : Ideal B)
/-- An `R`-algebra `A` is formally unramified if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at most one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyUnramified : Prop where
comp_injective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Injective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_unramified Algebra.FormallyUnramified
/-- An `R` algebra `A` is formally smooth if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists at least one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallySmooth : Prop where
comp_surjective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Surjective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_smooth Algebra.FormallySmooth
/-- An `R` algebra `A` is formally étale if for every `R`-algebra, every square-zero ideal
`I : Ideal B` and `f : A →ₐ[R] B ⧸ I`, there exists exactly one lift `A →ₐ[R] B`. -/
@[mk_iff]
class FormallyEtale : Prop where
comp_bijective :
∀ ⦃B : Type u⦄ [CommRing B],
∀ [Algebra R B] (I : Ideal B) (_ : I ^ 2 = ⊥),
Function.Bijective ((Ideal.Quotient.mkₐ R I).comp : (A →ₐ[R] B) → A →ₐ[R] B ⧸ I)
#align algebra.formally_etale Algebra.FormallyEtale
variable {R A}
theorem FormallyEtale.iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by
rw [FormallyUnramified_iff, FormallySmooth_iff, FormallyEtale_iff]
simp_rw [← forall_and]
rfl
#align algebra.formally_etale.iff_unramified_and_smooth Algebra.FormallyEtale.iff_unramified_and_smooth
instance (priority := 100) FormallyEtale.to_unramified [h : FormallyEtale R A] :
FormallyUnramified R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).1
#align algebra.formally_etale.to_unramified Algebra.FormallyEtale.to_unramified
instance (priority := 100) FormallyEtale.to_smooth [h : FormallyEtale R A] : FormallySmooth R A :=
(FormallyEtale.iff_unramified_and_smooth.mp h).2
#align algebra.formally_etale.to_smooth Algebra.FormallyEtale.to_smooth
theorem FormallyEtale.of_unramified_and_smooth [h₁ : FormallyUnramified R A]
[h₂ : FormallySmooth R A] : FormallyEtale R A :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨h₁, h₂⟩
#align algebra.formally_etale.of_unramified_and_smooth Algebra.FormallyEtale.of_unramified_and_smooth
theorem FormallyUnramified.lift_unique {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallyUnramified R A] (I : Ideal B) (hI : IsNilpotent I) (g₁ g₂ : A →ₐ[R] B)
(h : (Ideal.Quotient.mkₐ R I).comp g₁ = (Ideal.Quotient.mkₐ R I).comp g₂) : g₁ = g₂ := by
revert g₁ g₂
change Function.Injective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallyUnramified.comp_injective I hI
· intro B _ I J hIJ h₁ h₂ _ g₁ g₂ e
apply h₁
apply h₂
ext x
replace e := AlgHom.congr_fun e x
dsimp only [AlgHom.comp_apply, Ideal.Quotient.mkₐ_eq_mk] at e ⊢
rwa [Ideal.Quotient.eq, ← map_sub, Ideal.mem_quotient_iff_mem hIJ, ← Ideal.Quotient.eq]
#align algebra.formally_unramified.lift_unique Algebra.FormallyUnramified.lift_unique
theorem FormallyUnramified.ext [FormallyUnramified R A] (hI : IsNilpotent I) {g₁ g₂ : A →ₐ[R] B}
(H : ∀ x, Ideal.Quotient.mk I (g₁ x) = Ideal.Quotient.mk I (g₂ x)) : g₁ = g₂ :=
FormallyUnramified.lift_unique I hI g₁ g₂ (AlgHom.ext H)
#align algebra.formally_unramified.ext Algebra.FormallyUnramified.ext
theorem FormallyUnramified.lift_unique_of_ringHom [FormallyUnramified R A] {C : Type u} [CommRing C]
(f : B →+* C) (hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B)
(h : f.comp ↑g₁ = f.comp (g₂ : A →+* B)) : g₁ = g₂ :=
FormallyUnramified.lift_unique _ hf _ _
(by
ext x
have := RingHom.congr_fun h x
simpa only [Ideal.Quotient.eq, Function.comp_apply, AlgHom.coe_comp, Ideal.Quotient.mkₐ_eq_mk,
RingHom.mem_ker, map_sub, sub_eq_zero])
#align algebra.formally_unramified.lift_unique_of_ring_hom Algebra.FormallyUnramified.lift_unique_of_ringHom
theorem FormallyUnramified.ext' [FormallyUnramified R A] {C : Type u} [CommRing C] (f : B →+* C)
(hf : IsNilpotent <| RingHom.ker f) (g₁ g₂ : A →ₐ[R] B) (h : ∀ x, f (g₁ x) = f (g₂ x)) :
g₁ = g₂ :=
FormallyUnramified.lift_unique_of_ringHom f hf g₁ g₂ (RingHom.ext h)
#align algebra.formally_unramified.ext' Algebra.FormallyUnramified.ext'
theorem FormallyUnramified.lift_unique' [FormallyUnramified R A] {C : Type u} [CommRing C]
[Algebra R C] (f : B →ₐ[R] C) (hf : IsNilpotent <| RingHom.ker (f : B →+* C))
(g₁ g₂ : A →ₐ[R] B) (h : f.comp g₁ = f.comp g₂) : g₁ = g₂ :=
FormallyUnramified.ext' _ hf g₁ g₂ (AlgHom.congr_fun h)
#align algebra.formally_unramified.lift_unique' Algebra.FormallyUnramified.lift_unique'
theorem FormallySmooth.exists_lift {B : Type u} [CommRing B] [_RB : Algebra R B]
[FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I) (g : A →ₐ[R] B ⧸ I) :
∃ f : A →ₐ[R] B, (Ideal.Quotient.mkₐ R I).comp f = g := by
revert g
change Function.Surjective (Ideal.Quotient.mkₐ R I).comp
revert _RB
apply Ideal.IsNilpotent.induction_on (R := B) I hI
· intro B _ I hI _; exact FormallySmooth.comp_surjective I hI
· intro B _ I J hIJ h₁ h₂ _ g
let this : ((B ⧸ I) ⧸ J.map (Ideal.Quotient.mk I)) ≃ₐ[R] B ⧸ J :=
{
(DoubleQuot.quotQuotEquivQuotSup I J).trans
(Ideal.quotEquivOfEq (sup_eq_right.mpr hIJ)) with
commutes' := fun x => rfl }
obtain ⟨g', e⟩ := h₂ (this.symm.toAlgHom.comp g)
obtain ⟨g', rfl⟩ := h₁ g'
replace e := congr_arg this.toAlgHom.comp e
conv_rhs at e =>
rw [← AlgHom.comp_assoc, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_eq_coe,
AlgEquiv.comp_symm, AlgHom.id_comp]
exact ⟨g', e⟩
#align algebra.formally_smooth.exists_lift Algebra.FormallySmooth.exists_lift
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` square-zero,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : A →ₐ[R] B :=
(FormallySmooth.exists_lift I hI g).choose
#align algebra.formally_smooth.lift Algebra.FormallySmooth.lift
@[simp]
theorem FormallySmooth.comp_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) : (Ideal.Quotient.mkₐ R I).comp (FormallySmooth.lift I hI g) = g :=
(FormallySmooth.exists_lift I hI g).choose_spec
#align algebra.formally_smooth.comp_lift Algebra.FormallySmooth.comp_lift
@[simp]
theorem FormallySmooth.mk_lift [FormallySmooth R A] (I : Ideal B) (hI : IsNilpotent I)
(g : A →ₐ[R] B ⧸ I) (x : A) : Ideal.Quotient.mk I (FormallySmooth.lift I hI g x) = g x :=
AlgHom.congr_fun (FormallySmooth.comp_lift I hI g : _) x
#align algebra.formally_smooth.mk_lift Algebra.FormallySmooth.mk_lift
variable {C : Type u} [CommRing C] [Algebra R C]
/-- For a formally smooth `R`-algebra `A` and a map `f : A →ₐ[R] B ⧸ I` with `I` nilpotent,
this is an arbitrary lift `A →ₐ[R] B`. -/
noncomputable def FormallySmooth.liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C)
(g : B →ₐ[R] C) (hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
A →ₐ[R] B :=
FormallySmooth.lift _ hg' ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)
#align algebra.formally_smooth.lift_of_surjective Algebra.FormallySmooth.liftOfSurjective
@[simp]
theorem FormallySmooth.liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f)]
apply (Ideal.quotientKerAlgEquivOfSurjective hg).injective
rw [AlgEquiv.apply_symm_apply, Ideal.quotientKerAlgEquivOfSurjective,
Ideal.quotientKerAlgEquivOfRightInverse.apply]
exact (Ideal.kerLiftAlg_mk _ _).symm
#align algebra.formally_smooth.lift_of_surjective_apply Algebra.FormallySmooth.liftOfSurjective_apply
@[simp]
theorem FormallySmooth.comp_liftOfSurjective [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) :
g.comp (FormallySmooth.liftOfSurjective f g hg hg') = f :=
AlgHom.ext (FormallySmooth.liftOfSurjective_apply f g hg hg')
#align algebra.formally_smooth.comp_lift_of_surjective Algebra.FormallySmooth.comp_liftOfSurjective
end
section OfEquiv
variable {R : Type u} [CommSemiring R]
variable {A B : Type u} [Semiring A] [Algebra R A] [Semiring B] [Algebra R B]
theorem FormallySmooth.of_equiv [FormallySmooth R A] (e : A ≃ₐ[R] B) : FormallySmooth R B := by
constructor
intro C _ _ I hI f
use (FormallySmooth.lift I ⟨2, hI⟩ (f.comp e : A →ₐ[R] C ⧸ I)).comp e.symm
rw [← AlgHom.comp_assoc, FormallySmooth.comp_lift, AlgHom.comp_assoc, AlgEquiv.comp_symm,
AlgHom.comp_id]
#align algebra.formally_smooth.of_equiv Algebra.FormallySmooth.of_equiv
theorem FormallyUnramified.of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine' FormallyUnramified.comp_injective I hI _
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
#align algebra.formally_unramified.of_equiv Algebra.FormallyUnramified.of_equiv
theorem FormallyEtale.of_equiv [FormallyEtale R A] (e : A ≃ₐ[R] B) : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.of_equiv e, FormallySmooth.of_equiv e⟩
#align algebra.formally_etale.of_equiv Algebra.FormallyEtale.of_equiv
end OfEquiv
section Polynomial
open scoped Polynomial
variable (R : Type u) [CommSemiring R]
instance FormallySmooth.mvPolynomial (σ : Type u) : FormallySmooth R (MvPolynomial σ R) := by
constructor
intro C _ _ I _ f
have : ∀ s : σ, ∃ c : C, Ideal.Quotient.mk I c = f (MvPolynomial.X s) := fun s =>
Ideal.Quotient.mk_surjective _
choose g hg using this
refine' ⟨MvPolynomial.aeval g, _⟩
ext s
rw [← hg, AlgHom.comp_apply, MvPolynomial.aeval_X]
rfl
#align algebra.formally_smooth.mv_polynomial Algebra.FormallySmooth.mvPolynomial
instance FormallySmooth.polynomial : FormallySmooth R R[X] :=
FormallySmooth.of_equiv (MvPolynomial.pUnitAlgEquiv R)
#align algebra.formally_smooth.polynomial Algebra.FormallySmooth.polynomial
end Polynomial
section Comp
variable (R : Type u) [CommSemiring R]
variable (A : Type u) [CommSemiring A] [Algebra R A]
variable (B : Type u) [Semiring B] [Algebra R B] [Algebra A B] [IsScalarTower R A B]
theorem FormallySmooth.comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars R at e'
exact ⟨f''.restrictScalars _, e'.trans (AlgHom.ext fun _ => rfl)⟩
#align algebra.formally_smooth.comp Algebra.FormallySmooth.comp
theorem FormallyUnramified.comp [FormallyUnramified R A] [FormallyUnramified A B] :
FormallyUnramified R B := by
constructor
intro C _ _ I hI f₁ f₂ e
have e' :=
FormallyUnramified.lift_unique I ⟨2, hI⟩ (f₁.comp <| IsScalarTower.toAlgHom R A B)
(f₂.comp <| IsScalarTower.toAlgHom R A B) (by rw [← AlgHom.comp_assoc, e, AlgHom.comp_assoc])
letI := (f₁.comp (IsScalarTower.toAlgHom R A B)).toRingHom.toAlgebra
let F₁ : B →ₐ[A] C := { f₁ with commutes' := fun r => rfl }
let F₂ : B →ₐ[A] C := { f₂ with commutes' := AlgHom.congr_fun e'.symm }
ext1 x
change F₁ x = F₂ x
congr
exact FormallyUnramified.ext I ⟨2, hI⟩ (AlgHom.congr_fun e)
#align algebra.formally_unramified.comp Algebra.FormallyUnramified.comp
theorem FormallyUnramified.of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine' AlgHom.restrictScalars_injective R _
refine' FormallyUnramified.ext I ⟨2, e⟩ _
intro x
exact AlgHom.congr_fun e' x
#align algebra.formally_unramified.of_comp Algebra.FormallyUnramified.of_comp
theorem FormallyEtale.comp [FormallyEtale R A] [FormallyEtale A B] : FormallyEtale R B :=
FormallyEtale.iff_unramified_and_smooth.mpr
⟨FormallyUnramified.comp R A B, FormallySmooth.comp R A B⟩
#align algebra.formally_etale.comp Algebra.FormallyEtale.comp
end Comp
section OfSurjective
variable {R S : Type u} [CommRing R] [CommSemiring S]
variable {P A : Type u} [CommRing A] [Algebra R A] [CommRing P] [Algebra R P]
variable (I : Ideal P) (f : P →ₐ[R] A) (hf : Function.Surjective f)
theorem FormallySmooth.of_split [FormallySmooth R P] (g : A →ₐ[R] P ⧸ (RingHom.ker f.toRingHom) ^ 2)
(hg : f.kerSquareLift.comp g = AlgHom.id R A) : FormallySmooth R A := by
constructor
intro C _ _ I hI i
let l : P ⧸ (RingHom.ker f.toRingHom) ^ 2 →ₐ[R] C := by
refine' Ideal.Quotient.liftₐ _ (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) _
have : RingHom.ker f ≤ I.comap (FormallySmooth.lift I ⟨2, hI⟩ (i.comp f)) := by
rintro x (hx : f x = 0)
have : _ = i (f x) := (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x : _)
rwa [hx, map_zero, ← Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero] at this
intro x hx
have := (Ideal.pow_right_mono this 2).trans (Ideal.le_comap_pow _ 2) hx
rwa [hI] at this
have : i.comp f.kerSquareLift = (Ideal.Quotient.mkₐ R _).comp l := by
apply AlgHom.coe_ringHom_injective
apply Ideal.Quotient.ringHom_ext
ext x
exact (FormallySmooth.mk_lift I ⟨2, hI⟩ (i.comp f) x).symm
exact ⟨l.comp g, by rw [← AlgHom.comp_assoc, ← this, AlgHom.comp_assoc, hg, AlgHom.comp_id]⟩
#align algebra.formally_smooth.of_split Algebra.FormallySmooth.of_split
/-- Let `P →ₐ[R] A` be a surjection with kernel `J`, and `P` a formally smooth `R`-algebra,
then `A` is formally smooth over `R` iff the surjection `P ⧸ J ^ 2 →ₐ[R] A` has a section.
Geometric intuition: we require that a first-order thickening of `Spec A` inside `Spec P` admits
a retraction. -/
theorem FormallySmooth.iff_split_surjection [FormallySmooth R P] :
FormallySmooth R A ↔ ∃ g, f.kerSquareLift.comp g = AlgHom.id R A := by
constructor
· intro
have surj : Function.Surjective f.kerSquareLift := fun x =>
⟨Submodule.Quotient.mk (hf x).choose, (hf x).choose_spec⟩
have sqz : RingHom.ker f.kerSquareLift.toRingHom ^ 2 = 0 := by
rw [AlgHom.ker_kerSquareLift, Ideal.cotangentIdeal_square, Ideal.zero_eq_bot]
refine'
⟨FormallySmooth.lift _ ⟨2, sqz⟩ (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom,
_⟩
ext x
have :=
(Ideal.quotientKerAlgEquivOfSurjective surj).toAlgHom.congr_arg
(FormallySmooth.mk_lift _ ⟨2, sqz⟩
(Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom x)
-- Porting note: was
-- dsimp at this
-- rw [AlgEquiv.apply_symm_apply] at this
erw [AlgEquiv.apply_symm_apply] at this
conv_rhs => rw [← this, AlgHom.id_apply]
-- Porting note: lean3 was not finished here:
-- obtain ⟨y, e⟩ :=
-- Ideal.Quotient.mk_surjective
-- (FormallySmooth.lift _ ⟨2, sqz⟩
-- (Ideal.quotientKerAlgEquivOfSurjective surj).symm.toAlgHom
-- x)
-- dsimp at e ⊢
-- rw [← e]
-- rfl
· rintro ⟨g, hg⟩; exact FormallySmooth.of_split f g hg
#align algebra.formally_smooth.iff_split_surjection Algebra.FormallySmooth.iff_split_surjection
end OfSurjective
section UnramifiedDerivation
open scoped TensorProduct
variable {R S : Type u} [CommRing R] [CommRing S] [Algebra R S]
instance FormallyUnramified.subsingleton_kaehlerDifferential [FormallyUnramified R S] :
Subsingleton (Ω[S⁄R]) := by
rw [← not_nontrivial_iff_subsingleton]
intro h
obtain ⟨f₁, f₂, e⟩ := (KaehlerDifferential.endEquiv R S).injective.nontrivial
apply e
ext1
apply FormallyUnramified.lift_unique' _ _ _ _ (f₁.2.trans f₂.2.symm)
rw [← AlgHom.toRingHom_eq_coe, AlgHom.ker_kerSquareLift]
exact ⟨_, Ideal.cotangentIdeal_square _⟩
#align algebra.formally_unramified.subsingleton_kaehler_differential Algebra.FormallyUnramified.subsingleton_kaehlerDifferential
theorem FormallyUnramified.iff_subsingleton_kaehlerDifferential :
FormallyUnramified R S ↔ Subsingleton (Ω[S⁄R]) := by
constructor
· intros; infer_instance
· intro H
constructor
intro B _ _ I hI f₁ f₂ e
letI := f₁.toRingHom.toAlgebra
haveI := IsScalarTower.of_algebraMap_eq' f₁.comp_algebraMap.symm
have :=
((KaehlerDifferential.linearMapEquivDerivation R S).toEquiv.trans
(derivationToSquareZeroEquivLift I hI)).surjective.subsingleton
exact Subtype.ext_iff.mp (@Subsingleton.elim _ this ⟨f₁, rfl⟩ ⟨f₂, e.symm⟩)
#align algebra.formally_unramified.iff_subsingleton_kaehler_differential Algebra.FormallyUnramified.iff_subsingleton_kaehlerDifferential
end UnramifiedDerivation
section BaseChange
open scoped TensorProduct
variable {R : Type u} [CommSemiring R]
variable {A : Type u} [Semiring A] [Algebra R A]
variable (B : Type u) [CommSemiring B] [Algebra R B]
instance FormallyUnramified.base_change [FormallyUnramified R A] :
FormallyUnramified B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f₁ f₂ e
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
ext : 1
· exact Subsingleton.elim _ _
· exact FormallyUnramified.ext I ⟨2, hI⟩ fun x => AlgHom.congr_fun e (1 ⊗ₜ x)
#align algebra.formally_unramified.base_change Algebra.FormallyUnramified.base_change
instance FormallySmooth.base_change [FormallySmooth R A] : FormallySmooth B (B ⊗[R] A) := by
constructor
intro C _ _ I hI f
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine' ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) _, _⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp TensorProduct.includeRight)
· apply AlgHom.restrictScalars_injective R
apply TensorProduct.ext'
intro b a
suffices algebraMap B _ b * f (1 ⊗ₜ[R] a) = f (b ⊗ₜ[R] a) by simpa [Algebra.ofId_apply]
rw [← Algebra.smul_def, ← map_smul, TensorProduct.smul_tmul', smul_eq_mul, mul_one]
#align algebra.formally_smooth.base_change Algebra.FormallySmooth.base_change
instance FormallyEtale.base_change [FormallyEtale R A] : FormallyEtale B (B ⊗[R] A) :=
FormallyEtale.iff_unramified_and_smooth.mpr ⟨inferInstance, inferInstance⟩
#align algebra.formally_etale.base_change Algebra.FormallyEtale.base_change
end BaseChange
section Localization
variable {R S Rₘ Sₘ : Type u} [CommRing R] [CommRing S] [CommRing Rₘ] [CommRing Sₘ]
variable (M : Submonoid R)
variable [Algebra R S] [Algebra R Sₘ] [Algebra S Sₘ] [Algebra R Rₘ] [Algebra Rₘ Sₘ]
variable [IsScalarTower R Rₘ Sₘ] [IsScalarTower R S Sₘ]
variable [IsLocalization M Rₘ] [IsLocalization (M.map (algebraMap R S)) Sₘ]
-- Porting note: no longer supported
-- attribute [local elab_as_elim] Ideal.IsNilpotent.induction_on
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
use this
apply AlgHom.coe_ringHom_injective
|
refine' IsLocalization.ringHom_ext M _
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ := by
constructor
intro Q _ _ I e f
have : ∀ x : M, IsUnit (algebraMap R Q x) := by
intro x
apply (IsNilpotent.isUnit_quotient_mk_iff ⟨2, e⟩).mp
convert (IsLocalization.map_units Rₘ x).map f
simp only [Ideal.Quotient.mk_algebraMap, AlgHom.commutes]
let this : Rₘ →ₐ[R] Q :=
{ IsLocalization.lift this with commutes' := IsLocalization.lift_eq this }
use this
apply AlgHom.coe_ringHom_injective
|
Mathlib.RingTheory.Etale.482_0.sEffwLG8zJBnQIt
|
theorem FormallySmooth.of_isLocalization : FormallySmooth R Rₘ
|
Mathlib_RingTheory_Etale
|
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