75.15 Generating derived… to 75.29 Detecting Boundedness
Section75.15 Generating derived categories#
Section 75.15 of the Stacks Project, Tag 09IU. 6 tags here are carried by 12 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 09IV, a lemma
Tag 09IV in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_retract_restrict_of_supported, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectSupportedExtension.lean, line 673.
Stacks 09IV. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS,W ⊆ Xa quasi-compact open subspace andT ⊆ |X|a closed subset such thatX \ T ⟶ Xis quasi-compact. LetEbe an object ofD_QCoh(𝒪_X)andα : P ⟶ E|_Wa map wherePis a perfect object ofD(𝒪_W)supported onT ∩ W. Then there exists a mapβ : R ⟶ EwhereRis a perfect object ofD(𝒪_X)supported onTsuch thatPis a direct summand ofR|_WinD(𝒪_W)compatibly withαandβ|_W: there ares : P ⟶ R|_Wandr : R|_W ⟶ Pwiths ≫ r = 𝟙 Pands ≫ β|_W = α.
theorem exists_isPerfect_retract_restrict_of_supported
(hZ : B.Contains (⊤_ Scheme.{u})) (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1) (hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.1)
(W : Subfunctor X.1) (hW : IsOpenSubspace B S hS W)
(hWqc : IsQuasiCompactSpace B S hS W.toFunctor) (T : Set (points B S hS X.1))
(hT : IsClosed T)
(hTqc : IsQuasiCompact B S hS
((openSubspacesOrderIso B S hS X.1 X.2).symm ⟨Tᶜ, hT.isOpen_compl⟩).1.ι)
(E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2)
(P : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(hP : (smallEtaleRingedSite B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2).IsPerfect P)
(hPT : CohomologySupportedOn B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2 (pointsMap B S hS W.toFunctor W.ι ⁻¹' T) P)
(α : P ⟶ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj E.obj) :
∃ (R : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)) (β : R ⟶ E.obj),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect R ∧
CohomologySupportedOn B S hS X.1 X.2 T R ∧
∃ (s : P ⟶ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R)
(r : (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R ⟶ P),
s ≫ r = 𝟙 P ∧
s ≫ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.map β = α
Tag 09IW, a remark
Tag 09IW in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_restrict_iso_biprod_isIso_homologyFunctor_map_inl, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectSupportedExtensionCohomology.lean, line 757. The attribute's comment: “first and second assertions: the shape of the restriction, and its top cohomology”.
Stacks 09IW, first two assertions. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS,W ⊆ Xa quasi-compact open subspace,T ⊆ |X|a closed subset such thatX \ T ⟶ Xis quasi-compact,Ean object ofD_QCoh(𝒪_X)andα : P ⟶ E|_Wa morphism from a perfect objectPofD(𝒪_W)supported onT ∩ W. The construction of Stacks 09IV givesβ : R ⟶ E, withRa perfect object ofD(𝒪_X)supported onT, such that
R|_W ≅ P ⊕ CwithC ≅ P^{⊕ n₁}[1] ⊕ ⋯ ⊕ P^{⊕ nₘ}[m]for somem ≥ 0andn_j ≥ 0(CategoryTheory.IsFiniteSumOfPositiveShifts),β|_WbeingαonPand zero onC;- the highest degree cohomology sheaf of
R|_Wis that ofP: wheneverHⁱ(P) = 0for alli > d, the inclusion of the summandPinducesHⁱ(P) ≅ Hⁱ(R|_W)for everyi ≥ d.Both derived categories are taken at arbitrary models; "supported on" is the condition of Stacks 08GF on every cohomology sheaf, and
D_QCoh(𝒪_X)is the condition of Stacks 071X.
theorem exists_isPerfect_restrict_iso_biprod_isIso_homologyFunctor_map_inl
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(W : Subfunctor X.1) (hW : IsOpenSubspace B S hS W)
(hWqc : IsQuasiCompactSpace B S hS W.toFunctor) (T : Set (points B S hS X.1))
(hT : IsClosed T)
(hTqc : IsQuasiCompact B S hS
((openSubspacesOrderIso B S hS X.1 X.2).symm ⟨Tᶜ, hT.isOpen_compl⟩).1.ι)
[hDX : HasDerivedCategory.{w} (Modules.{u, u} B S hS X.1 X.2)]
[hDW : HasDerivedCategory.{w'} (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2)]
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O)) E)
(P : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(hP : (smallEtaleRingedSite B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2).IsPerfect P)
(hPT : ∀ i : ℤ, supportOfSheaf B S hS (openSubspaceSpace B S hS W hW).1
((SheafOfModules.toSheaf.{u} _).obj ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2) i).obj P)) ⊆
pointsMap B S hS W.toFunctor W.ι ⁻¹' T)
(α : P ⟶ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj E) :
∃ (R : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)) (β : R ⟶ E),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect R ∧
(∀ i : ℤ, supportOfSheaf B S hS X.1
((SheafOfModules.toSheaf.{u} _).obj
((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) i).obj R)) ⊆ T) ∧
∃ (C : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(e : (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R ≅ P ⊞ C),
IsFiniteSumOfPositiveShifts P C ∧
(etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.map β =
e.hom ≫ biprod.fst ≫ α ∧
∀ d : ℤ, (∀ i : ℤ, d < i → IsZero ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS
(openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2) i).obj P)) →
∀ i : ℤ, d ≤ i → IsIso ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS
(openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2) i).map (biprod.inl ≫ e.inv))
AlgebraicGeometry.Space.exists_isPerfect_retract_restrict_isIso_homologyFunctor_map, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectSupportedExtensionCohomology.lean, line 821. The attribute's comment: “third assertion: equality of cohomology sheaves above any given degree”.
Stacks 09IW, third assertion. In the situation of Stacks 09IV —
Sa scheme,Xa quasi-compact and quasi-separated algebraic space overS,W ⊆ Xa quasi-compact open subspace,T ⊆ |X|closed withX \ T ⟶ Xquasi-compact,EinD_QCoh(𝒪_X)andα : P ⟶ E|_WwithPperfect inD(𝒪_W)supported onT ∩ W— one can achieve equality of the cohomology sheaves ofR|_WandPabove any given degree: for every integerNthere isβ : R ⟶ EwithRperfect inD(𝒪_X)and supported onT, andPa direct summand ofR|_Wcompatibly withαandβ|_W—s : P ⟶ R|_Wandr : R|_W ⟶ Pwiths ≫ r = 𝟙 Pands ≫ β|_W = α, indeedβ|_W = r ≫ α— such thatHⁱ(s) : Hⁱ(P) ⟶ Hⁱ(R|_W)is an isomorphism for everyi ≥ N.Both derived categories are taken at arbitrary models.
theorem exists_isPerfect_retract_restrict_isIso_homologyFunctor_map
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(W : Subfunctor X.1) (hW : IsOpenSubspace B S hS W)
(hWqc : IsQuasiCompactSpace B S hS W.toFunctor) (T : Set (points B S hS X.1))
(hT : IsClosed T)
(hTqc : IsQuasiCompact B S hS
((openSubspacesOrderIso B S hS X.1 X.2).symm ⟨Tᶜ, hT.isOpen_compl⟩).1.ι)
[hDX : HasDerivedCategory.{w} (Modules.{u, u} B S hS X.1 X.2)]
[hDW : HasDerivedCategory.{w'} (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2)]
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O)) E)
(P : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(hP : (smallEtaleRingedSite B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2).IsPerfect P)
(hPT : ∀ i : ℤ, supportOfSheaf B S hS (openSubspaceSpace B S hS W hW).1
((SheafOfModules.toSheaf.{u} _).obj ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2) i).obj P)) ⊆
pointsMap B S hS W.toFunctor W.ι ⁻¹' T)
(α : P ⟶ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj E) (N : ℤ) :
∃ (R : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)) (β : R ⟶ E),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect R ∧
(∀ i : ℤ, supportOfSheaf B S hS X.1
((SheafOfModules.toSheaf.{u} _).obj
((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) i).obj R)) ⊆ T) ∧
∃ (s : P ⟶ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R)
(r : (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R ⟶ P),
s ≫ r = 𝟙 P ∧
s ≫ (etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.map β = α ∧
(etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.map β = r ≫ α ∧
∀ i : ℤ, N ≤ i → IsIso ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS
(openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2) i).map s)
Tag 09IX, a lemma
Tag 09IX in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_retract_restrict, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectDirectSummandExtension.lean, line 187.
Stacks 09IX. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS,Wa quasi-compact open subspace ofXandPa perfect object ofD(𝒪_W). ThenPis a direct summand of the restrictionQ|_Wof a perfect objectQofD(𝒪_X): there ares : P ⟶ Q|_Wandr : Q|_W ⟶ Pwiths ≫ r = 𝟙.The induction of Stacks 09IT on the quasi-compact opens containing
W, withAlgebraicGeometry.Space.isSummandOfPerfectRestriction_stepas the inductive step.
theorem exists_isPerfect_retract_restrict (hZ : B.Contains (⊤_ Scheme.{u}))
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.1) (W : Subfunctor X.1)
(hW : IsOpenSubspace B S hS W) (hWqc : IsQuasiCompactSpace B S hS W.toFunctor)
(P : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(hP : (smallEtaleRingedSite B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2).IsPerfect P) :
∃ Q : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect Q ∧
∃ (s : P ⟶ (etaleModulePullback.{u, u} B S hS
(ObjectProperty.homMk W.ι : openSubspaceSpace B S hS W hW ⟶ X)
(isEtale_of_isOpenImmersion B S hS hW.2)).mapDerivedCategory.obj Q)
(r : (etaleModulePullback.{u, u} B S hS
(ObjectProperty.homMk W.ι : openSubspaceSpace B S hS W hW ⟶ X)
(isEtale_of_isOpenImmersion B S hS hW.2)).mapDerivedCategory.obj Q ⟶ P),
s ≫ r = 𝟙 P
AlgebraicGeometry.Space.exists_isPerfect_restrict_iso_biprod_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectDirectSummandRestriction.lean, line 72.
Stacks 09IX. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overSandW ⊆ Xa quasi-compact open subspace. LetPbe a perfect object ofD(𝒪_W). ThenPis a direct summand of the restriction of a perfect object ofD(𝒪_X): there are a perfectRinD(𝒪_X)and an objectCofD(𝒪_W)withR|_W ≅ P ⊕ C. Both derived categories are taken at arbitrary models; quasi-separation is absolute, over the final object of the base category of schemes.
theorem exists_isPerfect_restrict_iso_biprod_of_isPerfect
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(W : Subfunctor X.1) (hW : IsOpenSubspace B S hS W)
(hWqc : IsQuasiCompactSpace B S hS W.toFunctor)
[hDX : HasDerivedCategory.{w} (Modules.{u, u} B S hS X.1 X.2)]
[hDW : HasDerivedCategory.{w'} (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2)]
(P : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2))
(hP : (smallEtaleRingedSite B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2).IsPerfect P) :
∃ R : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect R ∧
∃ C : DerivedCategory (Modules.{u, u} B S hS (openSubspaceSpace B S hS W hW).1
(openSubspaceSpace B S hS W hW).2),
Nonempty ((etaleModulePullback.{u, u} B S hS (openSubspaceHom B S hS W hW)
(isEtale_openSubspaceHom B S hS W hW)).mapDerivedCategory.obj R ≅ P ⊞ C)
Tag 09IY, a theorem
Tag 09IY in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_generator, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectCompactQcqs.lean, line 171.
Stacks 09IY. Let
Xbe a quasi-compact and quasi-separated algebraic space overS. There is a perfect objectPofD(𝒪_X)such that an objectEofD_QCoh(𝒪_X)is zero if and only ifHom(P[n], E) = 0for alln ∈ ℤ.The compact weak generator of
D_QCoh(𝒪_X)ofAlgebraicGeometry.Space.exists_isCompactObject_isWeakTriangulatedGenerator_of_qcqsis perfect by Stacks 09M8.
theorem exists_isPerfect_generator (X : AlgebraicSpaces B S hS)
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1) :
∃ P : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P ∧
∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero E.obj ↔ ∀ n : ℤ, Subsingleton (P⟦n⟧ ⟶ E.obj)
AlgebraicGeometry.Space.exists_isPerfect_isZero_iff_forall_hom_shift_eq_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/QcqsPerfectGenerator.lean, line 48.
Stacks 09IY. Let
Sbe a scheme andXa quasi-compact and quasi-separated algebraic space overS. The categoryD_QCoh(𝒪_X)is generated by a single perfect object: there is a perfect objectPofD(𝒪_X)such that forEinD_QCoh(𝒪_X)the following are equivalent: (1)E = 0, (2)Hom(P[n], E) = 0for alln ∈ ℤ.
theorem exists_isPerfect_isZero_iff_forall_hom_shift_eq_zero (B : BigFppfSite.{u}) (S : Scheme.{u})
(hS : B.Contains S) (hZ : B.Contains (⊤_ Scheme.{u})) (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1) (hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.1) :
∃ P : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P ∧
∀ E, qcohDerivedProp B S hS X E → (IsZero E ↔ ∀ (n : ℤ) (g : P⟦n⟧ ⟶ E), g = 0)
Tag 0E4R, a remark
Tag 0E4R in the Stacks Project.
AlgebraicGeometry.Space.tfae_reflectsIsomorphisms_quasicoherentDerivedPushforward, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/QuasicoherentDerivedPushforwardGenerator.lean, line 59. The attribute's comment: “the equivalence of (1), (2) and (3)”.
Stacks 0E4R. Let
f : X ⟶ Ybe a quasi-compact and quasi-separated morphism of algebraic spaces andEa generator ofD_QCoh(𝒪_Y). The following are equivalent: 1. forKinD_QCoh(𝒪_X),Rf_*K = 0if and only ifK = 0; 2.Rf_* : D_QCoh(𝒪_X) ⥤ D_QCoh(𝒪_Y)reflects isomorphisms; 3.Lf^*Eis a generator ofD_QCoh(𝒪_X).
theorem tfae_reflectsIsomorphisms_quasicoherentDerivedPushforward (f : X ⟶ Y)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(E : QuasicoherentDerivedCategory.{u, u} B S hS Y.1 Y.2)
(hE : ∀ K : QuasicoherentDerivedCategory.{u, u} B S hS Y.1 Y.2,
IsZero K ↔ ∀ n : ℤ, Subsingleton (E⟦n⟧ ⟶ K)) :
List.TFAE [∀ K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero ((quasicoherentDerivedPushforward B S hS f hfqc hfqs).obj K) ↔ IsZero K,
(quasicoherentDerivedPushforward B S hS f hfqc hfqs).ReflectsIsomorphisms,
∀ K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero K ↔ ∀ n : ℤ, Subsingleton (((quasicoherentDerivedModulePullback f).obj E)⟦n⟧ ⟶ K)]
AlgebraicGeometry.Space.isZero_quasicoherentDerivedPushforward_obj_iff_of_isAffineHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/QuasicoherentDerivedPushforwardGenerator.lean, line 75. The attribute's comment: “the conditions hold if f is affine”.
Stacks 0E4R, the case of an affine morphism. For
faffine,Rf_*K = 0if and only ifK = 0, forKinD_QCoh(𝒪_X).
theorem isZero_quasicoherentDerivedPushforward_obj_iff_of_isAffineHom (f : X ⟶ Y)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hf : IsAffineHom B S hS f.hom) (K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
IsZero ((quasicoherentDerivedPushforward B S hS f hfqc hfqs).obj K) ↔ IsZero K
AlgebraicGeometry.Space.isZero_quasicoherentDerivedPushforward_obj_iff_of_isOpenImmersion, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/QuasicoherentDerivedPushforwardGenerator.lean, line 86. The attribute's comment: “the conditions hold if f is an open immersion”.
Stacks 0E4R, the case of an open immersion. For an open immersion
j,Rj_*K = 0if and only ifK = 0: the restriction ofRj_*Kto the open isK.
theorem isZero_quasicoherentDerivedPushforward_obj_iff_of_isOpenImmersion (f : X ⟶ Y)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hf : IsOpenImmersion B S hS f.hom)
(K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
IsZero ((quasicoherentDerivedPushforward B S hS f hfqc hfqs).obj K) ↔ IsZero K
AlgebraicGeometry.Space.isZero_quasicoherentDerivedPushforward_obj_iff_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/QuasicoherentDerivedPushforwardGenerator.lean, line 103. The attribute's comment: “the conditions are stable under composition”.
Stacks 0E4R, compositions. If
Rf_*andRg_*detect zero objects ofD_QCoh, then so doesR(f ≫ g)_*.
theorem isZero_quasicoherentDerivedPushforward_obj_iff_comp (f : X ⟶ Y) (g : Y ⟶ Z)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hgqc : IsQuasiCompact B S hS g.hom) (hgqs : IsQuasiSeparated B S hS g.hom)
(hfgqc : IsQuasiCompact B S hS (f ≫ g).hom) (hfgqs : IsQuasiSeparated B S hS (f ≫ g).hom)
(hf : ∀ K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero ((quasicoherentDerivedPushforward B S hS f hfqc hfqs).obj K) ↔ IsZero K)
(hg : ∀ K : QuasicoherentDerivedCategory.{u, u} B S hS Y.1 Y.2,
IsZero ((quasicoherentDerivedPushforward B S hS g hgqc hgqs).obj K) ↔ IsZero K)
(K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
IsZero ((quasicoherentDerivedPushforward B S hS (f ≫ g) hfgqc hfgqs).obj K) ↔ IsZero K
Tag 0AEC, a lemma
Tag 0AEC in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_generator_of_supported, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SupportedPerfectGenerator.lean, line 393.
Stacks 0AEC. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS, andT ⊆ |X|a closed subset such that|X| ∖ Tis quasi-compact. ThenD_{QCoh,T}(𝒪_X)is generated by a single perfect object: there is a perfect objectPofD(𝒪_X)with quasi-coherent cohomology supported onT(an object ofD_{QCoh,T}(𝒪_X)) such that an objectEofD_QCoh(𝒪_X)supported onTis zero if and only ifHom(P[n], E) = 0for alln ∈ ℤ.
D_{QCoh,T}(𝒪_X)is the full subcategory ofD_QCoh(𝒪_X)of the objects whose cohomology sheaves are supported onT(AlgebraicGeometry.Space.CohomologySupportedOn, Stacks 08GF);|X| ∖ Tcarries the topology of|X|.
theorem exists_isPerfect_generator_of_supported (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(T : Set (points B S hS X.1)) (hT : IsClosed T) (hTc : IsCompact Tᶜ) :
∃ P : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P ∧
DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
((Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))) P ∧
CohomologySupportedOn B S hS X.1 X.2 T P ∧
∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
CohomologySupportedOn B S hS X.1 X.2 T E.obj →
(IsZero E.obj ↔ ∀ (n : ℤ) (g : P⟦n⟧ ⟶ E.obj), g = 0)
Sources
Section75.16 Compact and perfect objects#
Section 75.16 of the Stacks Project, Tag 09M7. 5 tags here are carried by 6 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 09M8, a proposition
Tag 09M8 in the Stacks Project.
AlgebraicGeometry.Space.isCompactObject_iff_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectCompactQcqs.lean, line 156.
Stacks 09M8. Let
Xbe a quasi-compact and quasi-separated algebraic space overS. An object ofD_QCoh(𝒪_X)is compact if and only if it is perfect.
theorem isCompactObject_iff_isPerfect (X : AlgebraicSpaces B S hS)
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
IsCompactObject.{u} E ↔ (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E.obj
Tag 0GFC, a remark
Tag 0GFC in the Stacks Project.
AlgebraicGeometry.Space.triangEnvelope_eq_isPerfect_of_isWeakTriangulatedGenerator, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectClassicalGenerator.lean, line 87.
Stacks 0GFC. Let
Xbe a quasi-compact and quasi-separated algebraic space overS, and letGbe a perfect object ofD(𝒪_X)which is a generator ofD_QCoh(𝒪_X). ThenGis a classical generator ofD_perf(𝒪_X): the triangulated envelope ofGinD_QCoh(𝒪_X)— its closure under shifts, finite direct sums, extensions and direct summands — is exactly the property of being perfect.The source combines [Stacks, Tag 08F3] (direct sums in
D_QCoh(𝒪_X)), [Stacks, Tag 09M8] (compact equals perfect) and [Stacks, Tag 09SR] (a compact generator classically generates the compact objects). Such aGexists by [Stacks, Tag 09IY]; seeAlgebraicGeometry.Space.exists_isPerfect_isWeakTriangulatedGenerator_triangEnvelope_eq.
theorem triangEnvelope_eq_isPerfect_of_isWeakTriangulatedGenerator (X : AlgebraicSpaces B S hS)
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(G : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2)
(hG : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect G.obj)
(hgen : (ObjectProperty.singleton G).IsWeakTriangulatedGenerator) :
(ObjectProperty.singleton G).triangEnvelope =
fun E => (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E.obj
AlgebraicGeometry.Space.exists_isPerfect_isWeakTriangulatedGenerator_triangEnvelope_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectClassicalGenerator.lean, line 109. The attribute's comment: “the existence of a perfect generator G of D_QCoh(O_X), by 09IY”.
Stacks 0GFC, the existence clause. On a quasi-compact and quasi-separated algebraic space
XoverSthere is a perfect objectGofD(𝒪_X)which is a generator ofD_QCoh(𝒪_X)([Stacks, Tag 09IY]), and every suchGis a classical generator ofD_perf(𝒪_X): its triangulated envelope inD_QCoh(𝒪_X)is the property of being perfect. The generator is the compact generator ofexists_isCompactObject_isWeakTriangulatedGenerator_of_qcqs, perfect by [Stacks, Tag 09M8], exactly as in the proof ofexists_isPerfect_generator.
theorem exists_isPerfect_isWeakTriangulatedGenerator_triangEnvelope_eq
(X : AlgebraicSpaces B S hS) (hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1) :
∃ G : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect G.obj ∧
(ObjectProperty.singleton G).IsWeakTriangulatedGenerator ∧
(ObjectProperty.singleton G).triangEnvelope =
fun E => (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E.obj
Tag 0AED, a lemma
Tag 0AED in the Stacks Project.
AlgebraicGeometry.Space.isCompactObject_iff_isPerfect_of_supported, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SupportedCompactPerfect.lean, line 77.
Stacks 0AED. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS, andT ⊆ |X|a closed subset such that|X| ∖ Tis quasi-compact. An object ofD_{QCoh,T}(𝒪_X)is compact (inD_{QCoh,T}(𝒪_X)) if and only if it is perfect as an object ofD(𝒪_X).
D_{QCoh,T}(𝒪_X)is the full subcategory ofD_QCoh(𝒪_X)of the objects whose cohomology sheaves are supported onT(AlgebraicGeometry.Space.quasicoherentDerivedSupportedOn, with support as in Stacks 08GF); compactness isCategoryTheory.IsCompactObject(Stacks 07LS) in that category, for direct sums indexed by the universeuof the site.
theorem isCompactObject_iff_isPerfect_of_supported (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(T : Set (points B S hS X.1)) (hT : IsClosed T) (hTc : IsCompact Tᶜ)
(E : (quasicoherentDerivedSupportedOn X T).FullSubcategory) :
IsCompactObject.{u} E ↔ (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E.obj.obj
Tag 0GFD, a remark
Tag 0GFD in the Stacks Project.
AlgebraicGeometry.Space.triangEnvelope_singleton_eq_isPerfect_of_supported, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SupportedPerfectClassicalGenerator.lean, line 71.
Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS, andT ⊆ |X|a closed subset such that|X| ∖ Tis quasi-compact. LetGbe a perfect object ofD(𝒪_X)lying inD_{QCoh,T}(𝒪_X)(quasi-coherent cohomology sheaves supported onT) which is a generator ofD_{QCoh,T}(𝒪_X): an objectEofD_QCoh(𝒪_X)supported onTwithHom(G[n], E) = 0for alln ∈ ℤis zero. ThenGis a classical generator ofD_{perf,T}(𝒪_X): the triangulated envelope ofGinD_{QCoh,T}(𝒪_X)— the smallest strictly full, saturated, triangulated subcategory containingG— is exactly the property of being perfect inD(𝒪_X).
theorem triangEnvelope_singleton_eq_isPerfect_of_supported (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(T : Set (points B S hS X.1)) (hT : IsClosed T) (hTc : IsCompact Tᶜ)
(G : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hG : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect G)
(hGq : DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
((Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))) G)
(hGs : CohomologySupportedOn B S hS X.1 X.2 T G)
(hgen : ∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
CohomologySupportedOn B S hS X.1 X.2 T E.obj →
(∀ (n : ℤ) (g : G⟦n⟧ ⟶ E.obj), g = 0) → IsZero E.obj) :
(ObjectProperty.singleton
(⟨⟨G, hGq⟩, hGs⟩ : (quasicoherentDerivedSupportedOn X T).FullSubcategory)).triangEnvelope =
fun K => (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect K.obj.obj
Tag 0AEE, a lemma
Tag 0AEE in the Stacks Project.
AlgebraicGeometry.Space.exists_isPerfect_comp_tensor_eq_zero_of_cohomologySupportedOn, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SupportedPerfectAnnihilator.lean, line 182.
Stacks 0AEE. Let
Sbe a scheme,Xa quasi-compact and quasi-separated algebraic space overS, andT ⊆ |X|a closed subset whose complementUis quasi-compact. Letα : P ⟶ Ebe a morphism ofD_QCoh(𝒪_X)withEsupported onT, where either (1)Pis perfect, or (2)Pis pseudo-coherent andEis bounded below. Then there are a perfect complexIof𝒪_X-modules and a mapu : I ⟶ 𝒪_X[0]such thatu|_U : I|_U ⟶ 𝒪_U[0]is an isomorphism andI ⊗^L P ⟶ 𝒪_X[0] ⊗^L P ≅ P ⟶ Eis zero.
Uis the open subspace ofXwith|U| = |X| ∖ T, and(-)|_Uthe restriction along its inclusion;Esupported onTmeans that every cohomology sheaf ofEis supported onT(AlgebraicGeometry.Space.CohomologySupportedOn, Stacks 08GF);Ebounded below means that its cohomology sheaves vanish in all sufficiently negative degrees.
theorem exists_isPerfect_comp_tensor_eq_zero_of_cohomologySupportedOn
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(T : Set (points B S hS X.1)) (hT : IsClosed T) (hTc : IsCompact Tᶜ)
{P E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)}
(hPq : QcohDerived B S hS X P) (hEq : QcohDerived B S hS X E) (α : P ⟶ E)
(hEs : CohomologySupportedOn B S hS X.1 X.2 T E)
(h : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P ∨
((smallEtaleRingedSite B S hS X.1 X.2).IsPseudoCoherent P ∧
DerivedCategoryB.IsBoundedBelow E)) :
∃ (I : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)) (u : I ⟶ unitDerived X),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect I ∧
IsIso ((etaleModulePullback.{u, u} B S hS
(openSubspaceHom B S hS
((openSubspacesOrderIso B S hS X.1 X.2).symm ⟨Tᶜ, hT.isOpen_compl⟩).1
((openSubspacesOrderIso B S hS X.1 X.2).symm ⟨Tᶜ, hT.isOpen_compl⟩).2)
(isEtale_openSubspaceHom B S hS _ _)).mapDerivedCategory.map u) ∧
((smallEtaleDerivedTensorProduct B S hS X).tensor.map u).app P ≫
(smallEtaleDerivedTensorUnitLeftIso B S hS X P).hom ≫ α = 0
Sources
Section75.17 Derived categories as module categories#
Section 75.17 of the Stacks Project, Tag 09M9. 6 tags here are carried by 8 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 09M9, the section
Tag 09M9 in the Stacks Project.
AlgebraicGeometry.RingedSite.derivedCategoryB_tensorWithComplexDerived_obj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Modules/TensorWithComplexQuasicoherentImage.lean, line 322. The attribute's comment: “RESTRICTED: 75.17.1 only”.
Stacks 09MA, and with it the content of section 09M9 that this file delivers. "Let
Sbe a scheme,Xan algebraic space overS. LetKbe a complex ofO_X-modules whose cohomology sheaves are quasi-coherent. Let(E, d) = Hom(K, K)be the endomorphism differential graded algebra. Then the functor- ⊗_E^𝐋 K : D(E, d) ⟶ D(O_X)of Tag 09LX has image contained inD_QCoh(O_X)."The dévissage is the source's: a differential graded
E-module with property (P) is the homotopy colimit of the stages of its filtration, each stage carries a finite filtration whose graded pieces are direct sums of shifts ofE, andE[k] ⊗_E KisK[k]. Tag 0FQE packages exactly that reduction, so what has to be checked is that lying inD_QCoh(O)holds at every shift of the free rank-one module — which is the staged computationAlgebraicGeometry.RingedSite.derivedCategoryB_tensorWithComplexDerived_obj_regulartogether with the stability ofD_QCoh(O)under the translation — is inherited by direct sums, and is inherited by the third vertex of a distinguished triangle from the other two, the last because- ⊗_E^𝐋 Kis exact andD_QCoh(O)is a triangulated subcategory ofD(O).Three assumptions are upstream inputs, not pieces of the conclusion: that quasi-coherence is a weak Serre class and that
hqcsum, a direct sum of quasi-coherentO-modules is quasi-coherent (the two clauses of Tag 03M1, the first as a typeclass assumption so that instance search supplies it where this repository has it); andhtensorSum, that tensoring withKcarries a direct sum of differential gradedE-modules to a direct sum inD(O)(the differential graded half of which is Tag 09LV, formalized asAlgebraicGeometry.RingedSite.tensorWithComplexDG_preservesCoproducts). The hypothesishKis the source's own, that the cohomology sheaves ofKbe quasi-coherent, stated for the complexliftComplex X Kthat the functor tensors with.Exactness of
- ⊗_E^𝐋 K, which the two-out-of-three step needs, is not assumed. As a hypothesishexact, Tag 09LW read against the pretriangulated structure that is not the one in force here, it would be unsatisfiable and this statement vacuous. What discharges it isAlgebraicGeometry.RingedSite.tensorWithComplexDerived_signTwist_isTriangulated, the unconditional exactness of- ⊗_E^𝐋 Kcomposed with the sign-twisted identity ofD(O): that composite has the same action on objects and on morphisms, so its image of a distinguished triangle has the same three vertices, which is all this step uses of it.
theorem derivedCategoryB_tensorWithComplexDerived_obj
[ObjectProperty.IsWeakSerreClass (tensorModOQuasicoherent.{u, v, w, t, s} X)]
(hqcsum : ∀ {ι : Type (max u v w t s)} (Mf : ι → TensorModO.{u, v, w, t, s} X)
(c : Cofan Mf), IsColimit c →
(∀ i, tensorModOQuasicoherent.{u, v, w, t, s} X (Mf i)) →
tensorModOQuasicoherent.{u, v, w, t, s} X c.pt)
(htensorSum : ∀ {ι : Type (max u v w t s)} [DecidableEq ι]
(N : ι → DifferentialGradedModule.{0, max u v w t s, max u v w t s}
(Rep.{u, v, w, t, s, 0} X ℤ K).algebra),
IsColimit ((toHomotopyCategory (Rep.{u, v, w, t, s, 0} X ℤ K).algebra ⋙
tensorWithComplexToDerivedCategory.{u, v, w, t, s, 0} X ℤ K).mapCocone
(Cofan.mk (General.directSumObj N) (General.directSumInj N))))
(hK : tensorModOQuasicoherentDerived.{u, v, w, t, s} X
(DerivedCategory.Q.obj (liftComplex.{u, v, w, t, s} X K)))
(M : DifferentialGradedModule.derivedCategory.{0, max u v w t s, max u v w t s}
(Rep.{u, v, w, t, s, 0} X ℤ K).algebra) :
tensorModOQuasicoherentDerived.{u, v, w, t, s} X
((tensorWithComplexDerived.{u, v, w, t, s, 0} X ℤ K).obj M)
Tag 09MA, a lemma
Tag 09MA in the Stacks Project.
AlgebraicGeometry.Space.derivedCategoryB_tensorWithComplexDerived_obj_smallEtale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TensorWithComplexQuasicoherentSmallEtale.lean, line 98.
Stacks 09MA. Let
Sbe a scheme, letFbe an algebraic space overS, letKbe a complex of𝒪_F-modules whose cohomology sheaves are quasi-coherent and let(E, d) = Hom(K, K)be the endomorphism differential graded algebra. Then the functor- ⊗_E^𝐋 K : D(E, d) ⟶ D(𝒪_F)of Stacks 09LX has image contained inD_QCoh(𝒪_F).The hypothesis
hKis the source's, that the cohomology sheaves ofKbe quasi-coherent, stated for the complexliftComplex F Kthat the functor tensors with. Nothing else is assumed: the weak Serre class that the dévissageAlgebraicGeometry.RingedSite.derivedCategoryB_tensorWithComplexDerived_obj_of_hqcsumuses comes fromAlgebraicGeometry.Space.quasicoherentSheaves_isWeakSerreClasstogether withAlgebraicGeometry.Space.isClosedUnderKernels,AlgebraicGeometry.Space.isClosedUnderCokernelsandAlgebraicGeometry.Space.isClosedUnderExtensionsat the module universe in which the tensor products are formed, and itshqcsumisAlgebraicGeometry.Space.isClosedUnderDirectSumsQuasicoherent; the exactness of the functor and its preservation of direct sums are instances of this development.
public theorem derivedCategoryB_tensorWithComplexDerived_obj_smallEtale
(K : CochainComplex (ModO.{u + 1, u, u, t} (smallEtaleRingedSite B S hS F hF)) ℤ)
(hK : tensorModOQuasicoherentDerived.{u + 1, u, u, t, s}
(smallEtaleRingedSite B S hS F hF)
(DerivedCategory.Q.obj (liftComplex.{u + 1, u, u, t, s}
(smallEtaleRingedSite B S hS F hF) K)))
(M : DifferentialGradedModule.derivedCategory.{0, max (u + 1) t s, max (u + 1) t s}
(RingedSite.Rep.{u + 1, u, u, t, s, 0} (smallEtaleRingedSite B S hS F hF) ℤ K).algebra) :
tensorModOQuasicoherentDerived.{u + 1, u, u, t, s} (smallEtaleRingedSite B S hS F hF)
((tensorWithComplexDerived.{u + 1, u, u, t, s, 0}
(smallEtaleRingedSite B S hS F hF) ℤ K).obj M)
AlgebraicGeometry.RingedSite.derivedCategoryB_tensorWithComplexDerived_obj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Modules/TensorWithComplexQuasicoherentImage.lean, line 322. The attribute's comment: “RESTRICTED: assumes 03M1, 09LV”.
Stacks 09MA, and with it the content of section 09M9 that this file delivers. "Let
Sbe a scheme,Xan algebraic space overS. LetKbe a complex ofO_X-modules whose cohomology sheaves are quasi-coherent. Let(E, d) = Hom(K, K)be the endomorphism differential graded algebra. Then the functor- ⊗_E^𝐋 K : D(E, d) ⟶ D(O_X)of Tag 09LX has image contained inD_QCoh(O_X)."The dévissage is the source's: a differential graded
E-module with property (P) is the homotopy colimit of the stages of its filtration, each stage carries a finite filtration whose graded pieces are direct sums of shifts ofE, andE[k] ⊗_E KisK[k]. Tag 0FQE packages exactly that reduction, so what has to be checked is that lying inD_QCoh(O)holds at every shift of the free rank-one module — which is the staged computationAlgebraicGeometry.RingedSite.derivedCategoryB_tensorWithComplexDerived_obj_regulartogether with the stability ofD_QCoh(O)under the translation — is inherited by direct sums, and is inherited by the third vertex of a distinguished triangle from the other two, the last because- ⊗_E^𝐋 Kis exact andD_QCoh(O)is a triangulated subcategory ofD(O).Three assumptions are upstream inputs, not pieces of the conclusion: that quasi-coherence is a weak Serre class and that
hqcsum, a direct sum of quasi-coherentO-modules is quasi-coherent (the two clauses of Tag 03M1, the first as a typeclass assumption so that instance search supplies it where this repository has it); andhtensorSum, that tensoring withKcarries a direct sum of differential gradedE-modules to a direct sum inD(O)(the differential graded half of which is Tag 09LV, formalized asAlgebraicGeometry.RingedSite.tensorWithComplexDG_preservesCoproducts). The hypothesishKis the source's own, that the cohomology sheaves ofKbe quasi-coherent, stated for the complexliftComplex X Kthat the functor tensors with.Exactness of
- ⊗_E^𝐋 K, which the two-out-of-three step needs, is not assumed. As a hypothesishexact, Tag 09LW read against the pretriangulated structure that is not the one in force here, it would be unsatisfiable and this statement vacuous. What discharges it isAlgebraicGeometry.RingedSite.tensorWithComplexDerived_signTwist_isTriangulated, the unconditional exactness of- ⊗_E^𝐋 Kcomposed with the sign-twisted identity ofD(O): that composite has the same action on objects and on morphisms, so its image of a distinguished triangle has the same three vertices, which is all this step uses of it.
theorem derivedCategoryB_tensorWithComplexDerived_obj
[ObjectProperty.IsWeakSerreClass (tensorModOQuasicoherent.{u, v, w, t, s} X)]
(hqcsum : ∀ {ι : Type (max u v w t s)} (Mf : ι → TensorModO.{u, v, w, t, s} X)
(c : Cofan Mf), IsColimit c →
(∀ i, tensorModOQuasicoherent.{u, v, w, t, s} X (Mf i)) →
tensorModOQuasicoherent.{u, v, w, t, s} X c.pt)
(htensorSum : ∀ {ι : Type (max u v w t s)} [DecidableEq ι]
(N : ι → DifferentialGradedModule.{0, max u v w t s, max u v w t s}
(Rep.{u, v, w, t, s, 0} X ℤ K).algebra),
IsColimit ((toHomotopyCategory (Rep.{u, v, w, t, s, 0} X ℤ K).algebra ⋙
tensorWithComplexToDerivedCategory.{u, v, w, t, s, 0} X ℤ K).mapCocone
(Cofan.mk (General.directSumObj N) (General.directSumInj N))))
(hK : tensorModOQuasicoherentDerived.{u, v, w, t, s} X
(DerivedCategory.Q.obj (liftComplex.{u, v, w, t, s} X K)))
(M : DifferentialGradedModule.derivedCategory.{0, max u v w t s, max u v w t s}
(Rep.{u, v, w, t, s, 0} X ℤ K).algebra) :
tensorModOQuasicoherentDerived.{u, v, w, t, s} X
((tensorWithComplexDerived.{u, v, w, t, s, 0} X ℤ K).obj M)
Tag 09MB, a lemma
Tag 09MB in the Stacks Project.
AlgebraicGeometry.Space.exists_window_subsingleton_hom_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectHomVanishingWindow.lean, line 95. The attribute's comment: “part (1)”.
Stacks 09MB, part (1). Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overSand letKbe a perfect object ofD(𝒪_X). Then there are integersa ≤ bsuch thatHom_{D(𝒪_X)}(K, L) = 0for everyL ∈ D_QCoh(𝒪_X)withHⁱ(L) = 0fori ∈ [a, b].The model of the derived category is arbitrary. The window is the one produced at the standard model by the induction
AlgebraicGeometry.Space.hasHomWindow_of_isPerfect_of_affine, whose affine case isAlgebraicGeometry.Space.hasHomWindow_of_isAffineSpace, carried across the comparison of the two models.
theorem exists_window_subsingleton_hom_of_isPerfect
(hqc : IsQuasiCompactSpace B S hS X) (hqs : IsAbsolutelyQuasiSeparated B S hS hZ X)
[hD : HasDerivedCategory.{w} (Modules.{u, u} B S hS X hX)]
(K : DerivedCategory (Modules.{u, u} B S hS X hX))
(hK : (smallEtaleRingedSite B S hS X hX).IsPerfect K) :
∃ a b : ℤ, a ≤ b ∧ ∀ L : DerivedCategory (Modules.{u, u} B S hS X hX),
(∀ i : ℤ, ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X hX) i).obj
L).IsQuasicoherentOnSite) →
(∀ i : ℤ, a ≤ i → i ≤ b →
IsZero ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X hX) i).obj L)) →
Subsingleton (K ⟶ L)
AlgebraicGeometry.Space.finite_setOf_not_subsingleton_hom_shift_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectHomVanishingWindow.lean, line 135. The attribute's comment: “part (2)”.
Stacks 09MB, part (2). Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overS, letKbe a perfect object ofD(𝒪_X)and letLbe a bounded object ofD_QCoh(𝒪_X). ThenExtⁿ_{D(𝒪_X)}(K, L) = Hom_{D(𝒪_X)}(K, L⟦n⟧)is zero for all but finitely manyn.Outside a bounded range of
nthe shiftL⟦n⟧has no cohomology at all in the window of part (1), becauseLis bounded.
theorem finite_setOf_not_subsingleton_hom_shift_of_isPerfect
(hqc : IsQuasiCompactSpace B S hS X) (hqs : IsAbsolutelyQuasiSeparated B S hS hZ X)
[HasDerivedCategory.{w} (Modules.{u, u} B S hS X hX)]
(K : DerivedCategory (Modules.{u, u} B S hS X hX))
(hK : (smallEtaleRingedSite B S hS X hX).IsPerfect K)
(L : DerivedCategory (Modules.{u, u} B S hS X hX))
(hLqc : ∀ i : ℤ, ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X hX) i).obj
L).IsQuasicoherentOnSite)
(c d : ℤ) (hLGE : L.IsGE c) (hLLE : L.IsLE d) :
{n : ℤ | ¬ Subsingleton (K ⟶ L⟦n⟧)}.Finite
Tag 09MC, a theorem
Tag 09MC in the Stacks Project.
AlgebraicGeometry.Space.exists_differentialGradedAlgebra_equivalence, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedModuleCategory.lean, line 266.
Stacks 09MC. Let
Xbe a quasi-compact and quasi-separated algebraic space overS. Then there exists a differential graded algebra(E, d)with only a finite number of nonzero cohomology groupsH^i(E)such thatD_QCoh(𝒪_X)is equivalent toD(E, d).
theorem exists_differentialGradedAlgebra_equivalence (hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1) :
∃ E : DifferentialGradedAlgebra.{0, u} ℤ, {i : ℤ | Nontrivial (E.Cohomology i)}.Finite ∧
Nonempty (QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2 ≌
DifferentialGradedModule.derivedCategory.{0, u, u} E)
Tag 0DK8, a remark
Tag 0DK8 in the Stacks Project.
AlgebraicGeometry.Space.exists_differentialGradedAlgebra_equivalence_of_supported, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedModuleCategorySupported.lean, line 199.
Stacks 0DK8. Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overSand letT ⊆ |X|be a closed subset such that|X| ∖ Tis quasi-compact. Then there exists a differential graded algebra(E, d)with only a finite number of nonzero cohomology groupsH^i(E)such thatD_{QCoh,T}(𝒪_X)is equivalent toD(E, d).This is the analogue of Stacks 09MC for
D_{QCoh,T}(𝒪_X), the full subcategory(AlgebraicGeometry.Space.quasicoherentDerivedSupportedOn X T).FullSubcategoryofD_QCoh(𝒪_X)of the objects whose cohomology sheaves are supported onT; forT = |X|it is Stacks 09MC.Eis the endomorphism differential graded algebra of a K-injective complex representing a perfect generator ofD_{QCoh,T}(𝒪_X)(Stacks 0AEC), formed on the equi-universe model of the small étale ringed site ofX, so that its carrier andD(E, d)live in the universe ofX.
theorem exists_differentialGradedAlgebra_equivalence_of_supported
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(hT : IsClosed T) (hTc : IsCompact Tᶜ) :
∃ E : DifferentialGradedAlgebra.{0, u} ℤ, {i : ℤ | Nontrivial (E.Cohomology i)}.Finite ∧
Nonempty ((quasicoherentDerivedSupportedOn X T).FullSubcategory ≌
DifferentialGradedModule.derivedCategory.{0, u, u} E)
Tag 0DK9, a remark
Tag 0DK9 in the Stacks Project.
AlgebraicGeometry.Space.exists_equivalence_derivedTensor_complexHomBimodule, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DifferentialGradedAlgebraUniqueness.lean, line 196.
Stacks 0DK9. Let
Xbe a quasi-compact and quasi-separated algebraic space over a ringR, and letKandLbe K-injective complexes of𝒪_X-modules corresponding to perfect generators ofD_QCoh(𝒪_X), withA = Hom_{Comp^dg(𝒪_X)}(K, K)andB = Hom_{Comp^dg(𝒪_X)}(L, L). SetΩ = Hom_{Comp^dg(𝒪_X)}(K, L)andΩ' = Hom_{Comp^dg(𝒪_X)}(L, K). ThenΩis a differential gradedB^opp ⊗_R A-module andΩ'a differential gradedA^opp ⊗_R B-module, and the equivalenceD(A, d) ⥤ D_QCoh(𝒪_X) ⥤ D(B, d)is given by- ⊗_A^𝐋 Ω', and its quasi-inverse by- ⊗_B^𝐋 Ω.The bimodules are
AlgebraicGeometry.RingedSite.complexHomBimodule:Ω' = Hom(L, K)is aDifferentialGraded.Bimodule A B(leftA-action by postcomposition, rightB-action by precomposition) andΩ = Hom(K, L)aDifferentialGraded.Bimodule B A. The equivalencesD(A, d) ≌ D_QCoh(𝒪_X)andD(B, d) ≌ D_QCoh(𝒪_X)are those of the proof of Stacks 09MC,AlgebraicGeometry.Space.dgaQuasicoherentEquivalence. The conclusion is an equivalenceD(A, d) ≌ D(B, d)whose functor is- ⊗_A^𝐋 Ω'and whose inverse is- ⊗_B^𝐋 Ω, isomorphic to the compositeD(A, d) ⥤ D_QCoh(𝒪_X) ⥤ D(B, d)and to its quasi-inverse: the situation of Stacks 09ST.
Xis overRthrough the ring mapψ : R ⟶ Γ(X, 𝒪_X);KandLare complexes on the equi-universe model of the small étale site ofX, corresponding throughAlgebraicGeometry.Space.modelEquivalenceto perfect generatorsPandQofD_QCoh(𝒪_X).
theorem exists_equivalence_derivedTensor_complexHomBimodule
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(K L : CochainComplex (RingedSite.ModO.{u, u, u, u} (modelSite X)) ℤ)
[(RingedSite.liftComplex.{u, u, u, u, u} (modelSite X) K).IsKInjective]
[(RingedSite.liftComplex.{u, u, u, u, u} (modelSite X) L).IsKInjective]
(P Q : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hP : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P)
(hQ : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect Q)
(hgenP : ∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero E.obj ↔ ∀ n : ℤ, Subsingleton (P⟦n⟧ ⟶ E.obj))
(hgenQ : ∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
IsZero E.obj ↔ ∀ n : ℤ, Subsingleton (Q⟦n⟧ ⟶ E.obj))
(eP : (modelEquivalence X).inverse.obj
(DerivedCategory.Q.obj (RingedSite.liftComplex.{u, u, u, u, u} (modelSite X) K) :
DerivedCategory (SheafOfModules.{u}
(Stacks.SheafOfModules.Tensor.ringSheaf (modelSite X).O))) ≅ P)
(eQ : (modelEquivalence X).inverse.obj
(DerivedCategory.Q.obj (RingedSite.liftComplex.{u, u, u, u, u} (modelSite X) L) :
DerivedCategory (SheafOfModules.{u}
(Stacks.SheafOfModules.Tensor.ringSheaf (modelSite X).O))) ≅ Q) :
letI := modelLinear X ψ
letI := modelScalarSections X ψ
∃ e : DifferentialGradedModule.derivedCategory.{u, u, u}
(RingedSite.Rep.{u, u, u, u, u, u} (modelSite X) R K).algebra ≌
DifferentialGradedModule.derivedCategory.{u, u, u}
(RingedSite.Rep.{u, u, u, u, u, u} (modelSite X) R L).algebra,
e.functor = DifferentialGraded.Bimodule.derivedTensorFunctor
(RingedSite.complexHomBimodule.{u, u, u, u, u} (modelSite X) R K L) ∧
e.inverse = DifferentialGraded.Bimodule.derivedTensorFunctor
(RingedSite.complexHomBimodule.{u, u, u, u, u} (modelSite X) R L K) ∧
Nonempty (e.functor ≅
((dgaQuasicoherentEquivalence X ψ K
(isRelativelyCompactGenerator_model hqc hqs K P hP hgenP eP)).trans
(dgaQuasicoherentEquivalence X ψ L
(isRelativelyCompactGenerator_model hqc hqs L Q hQ hgenQ eQ)).symm).functor) ∧
Nonempty (e.inverse ≅
((dgaQuasicoherentEquivalence X ψ K
(isRelativelyCompactGenerator_model hqc hqs K P hP hgenP eP)).trans
(dgaQuasicoherentEquivalence X ψ L
(isRelativelyCompactGenerator_model hqc hqs L Q hQ hgenQ eQ)).symm).inverse)
Sources
Section75.18 Characterizing pseudo-coherent complexes, I#
Section 75.18 of the Stacks Project, Tag 0DKA. 2 tags here are carried by 2 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0DKB, a lemma
Tag 0DKB in the Stacks Project.
AlgebraicGeometry.Space.isPseudoCoherent_iff_exists_isHomotopyColimitWith, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PseudoCoherentHomotopyColimit.lean, line 390.
Stacks 0DKB. Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overSand letK ∈ D(𝒪_X). The following are equivalent: 1.Kis pseudo-coherent; 2.K = hocolim KₙwhereKₙis perfect andτ_{≥-n}Kₙ ⟶ τ_{≥-n}Kis an isomorphism for alln.The model of
D(𝒪_X)is arbitrary. The system isF : ℕ ⥤ D(𝒪_X), indexed from0, its memberF.obj nbeing the source'sK_{n+1}; so the truncation condition at indexnis in degree-(n + 1). The mapsKₙ ⟶ Kare the coprojectionsc nof the derived colimit (CategoryTheory.Pretriangulated.IsHomotopyColimitWith), whose direct sum⊕ Kₙexists because the𝒪_X-modules form a Grothendieck abelian category.
theorem isPseudoCoherent_iff_exists_isHomotopyColimitWith
(hqc : IsQuasiCompactSpace B S hS X) (hqs : IsAbsolutelyQuasiSeparated B S hS hZ X)
[HasDerivedCategory.{w} (Modules.{u, u} B S hS X hX)]
(K : DerivedCategory (Modules.{u, u} B S hS X hX)) :
haveI := hasColimitsOfShape_discrete_nat_derivedCategory_modules B S hS X hX
(smallEtaleRingedSite B S hS X hX).IsPseudoCoherent K ↔
∃ (F : ℕ ⥤ DerivedCategory (Modules.{u, u} B S hS X hX)) (c : ∀ n, F.obj n ⟶ K),
(∀ n, (smallEtaleRingedSite B S hS X hX).IsPerfect (F.obj n)) ∧
(∀ n : ℕ,
IsIso ((DerivedCategory.TStructure.t.truncGE (-((n : ℤ) + 1))).map (c n))) ∧
IsHomotopyColimitWith F K c
Tag 0DKC, a lemma
Tag 0DKC in the Stacks Project.
AlgebraicGeometry.PerfectApproximation.isPseudoCoherent_iff_exists_isHomotopyColimitWith, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Perfect/PseudoCoherentComplex/HomotopyColimitSupported.lean, line 358.
[Stacks, Tag 0DKC]. Let
Xbe a quasi-compact and quasi-separated scheme, letT ⊆ Xbe a closed subset such thatX \ Tis quasi-compact, and letK ∈ D(𝒪_X)be supported onT. The following are equivalent: 1.Kis pseudo-coherent; 2.K = hocolim KₙwhereKₙis perfect, supported onT, andτ_{≥-n}Kₙ ⟶ τ_{≥-n}Kis an isomorphism for alln.The system is
F : ℕ ⥤ D(𝒪_X), indexed from0, its memberF.obj nbeing the source'sK_{n+1}; so the truncation condition at indexnis in degree-(n + 1). The mapsKₙ ⟶ Kare the coprojectionsc nof the derived colimit (CategoryTheory.Pretriangulated.IsHomotopyColimitWith). The same statement is [Stacks, Tag 0DJP].
theorem isPseudoCoherent_iff_exists_isHomotopyColimitWith [CompactSpace X]
[QuasiSeparatedSpace X] (T : Set X) (hT : IsClosed T) (hTc : IsCompact Tᶜ)
(K : DerivedCategory X.Modules)
(hK : Stacks.Cohomology.PushforwardRestriction.CohomologySupportedOn X.ringCatSheaf T K) :
RingedSpace.IsPseudoCoherent (X := X.toRingedSpace) K ↔
∃ (F : ℕ ⥤ DerivedCategory X.Modules) (c : ∀ n, F.obj n ⟶ K),
(∀ n, IsPerfect (X := X.toRingedSpace) (F.obj n)) ∧
(∀ n, Stacks.Cohomology.PushforwardRestriction.CohomologySupportedOn X.ringCatSheaf T
(F.obj n)) ∧
(∀ n : ℕ,
IsIso ((DerivedCategory.TStructure.t.truncGE (-((n : ℤ) + 1))).map (c n))) ∧
IsHomotopyColimitWith F K c
Sources
Section75.19 The coherator revisited#
Section 75.19 of the Stacks Project, Tag 0CR3. 5 tags here are carried by 8 declarations of the repository (all in StacksFrontier) and 1 registered declaration, quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0CR4, a lemma
Tag 0CR4 in the Stacks Project.
AlgebraicGeometry.Space.isLeftAdjoint_ι_derivedCategoryB_isQuasicoherentOnSite, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoherator.lean, line 365.
Stacks 0CR4. Let
Sbe a scheme andXa quasi-compact and quasi-separated algebraic space overS. The inclusion functorD_QCoh(𝒪_X) ⥤ D(𝒪_X)has a right adjoint, the coheratorDQ_X.This is the source's first proof: the induction principle of Stacks 08GP (
AlgebraicGeometry.Space.isLeftAdjoint_ι_of_forall_isAffineSpace) from the affine case (AlgebraicGeometry.Space.isLeftAdjoint_ι_of_isAffineSpace). The statement holds at every model of the derived categoryD(𝒪_X), by the model comparisonStacksDraft.DerivedModel.isLeftAdjoint_ι_derivedCategoryB_of_model.
theorem isLeftAdjoint_ι_derivedCategoryB_isQuasicoherentOnSite
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(h : HasDerivedCategory.{w} (Modules.{u, u} B S hS X.1 X.2)) :
(@DerivedCategoryB.derivedCategoryB _ _ _ h
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι.IsLeftAdjoint
AlgebraicGeometry.Space.quasicoherentDerivedInclusion_isLeftAdjoint_of_qcqs, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/QcqsCoherator.lean, line 109.
Stacks 0CR4. Let
Sbe a scheme and letXbe a quasi-compact and quasi-separated algebraic space overS. The inclusion functorD_QCoh(𝒪_X) ⥤ D(𝒪_X)has a right adjoint.The proof is the source's first one; see the module documentation. The conclusion is
CategoryTheory.Functor.IsLeftAdjoint, the existence of a right adjoint, which is what the tag asserts.
theorem quasicoherentDerivedInclusion_isLeftAdjoint_of_qcqs
(B : BigFppfSite.{u}) (S : Scheme.{u}) (hS : B.Contains S)
(hZ : B.Contains (⊤_ Scheme.{u}))
(F : (BigFppfOver B S hS)ᵒᵖ ⥤ Type u) (hF : IsAlgebraicSpace B S hS F)
(hFqc : IsQuasiCompactSpace B S hS F) (hFqs : IsAbsolutelyQuasiSeparated B S hS hZ F) :
(qcohDerivedProp B S hS ⟨F, hF⟩).ι.IsLeftAdjoint
Tag 0CR5, a lemma
Tag 0CR5 in the Stacks Project.
Declarations registered for the tag:
AlgebraicGeometry.Space.quasicoherentDerivedPushforwardCompCoheratorIso, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/QuasicoherentDerivedPushforwardTags.lean, line 29 (StacksFrontier).
Tag 0CR6, a remark
Tag 0CR6 in the Stacks Project.
AlgebraicGeometry.Space.coherator_edsMayerVietoris_triangle, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CoheratorMayerVietoris.lean, line 298.
Stacks 0CR6. Let
Sbe a scheme and(U ⊆ X, j_V : V ⟶ X)an elementary distinguished square of algebraic spaces overS(hereX = W,U = Uo,j_V = fV) withX,UandVquasi-compact and quasi-separated, and letP = U ×_X V.
- The functors
DQ_X,DQ_U,DQ_V,DQ_Pexist: the four inclusionsD_QCoh ⥤ Dhave right adjoints (Stacks 0CR4, at the four spaces;Pis quasi-compact and quasi-separated byAlgebraicGeometry.Space.isQuasiCompact_edsInterHomandAlgebraicGeometry.Space.isQuasiSeparated_edsInterHom).- The canonical triangle: for every choice of these right adjoints, with their adjunctions, and every
KinD(𝒪_X)there is a distinguished triangle
DQ_X(K) ⟶ Rj_{U,*}DQ_U(K|_U) ⊕ Rj_{V,*}DQ_V(K|_V) ⟶ Rj_{P,*}DQ_P(K|_P) ⟶ DQ_X(K)[1]whose two maps are the canonical ones: the exact functor
DQ_Xapplied to the two maps of the Mayer–Vietoris triangle of Stacks 08GW (AlgebraicGeometry.Space.exists_mem_distTriang_edsDerivedRestrictPushMayerVietorisLowerShriek, whose maps are written out here verbatim: the derived unitsK ⟶ R(j_{Y,*}(K|_Y))and the difference of the derived comparisonsR(j_{U,*}(K|_U)) ⟶ R(j_{P,*}(K|_P)),R(j_{V,*}(K|_V)) ⟶ R(j_{P,*}(K|_P))), conjugated by the three isomorphismsDQ_X(R(j_{Y,*}(K|_Y))) ≅ Rj_{Y,*}DQ_Y(K|_Y)(AlgebraicGeometry.Space.coheratorRestrictPushObjIso), which are Stacks 0CR5 for the étale morphismj_Ycomposed withDQ_Xof the canonical identificationR(j_{Y,*}((-)|_Y)) ≅ Rj_{Y,*}((-)|_Y). Only the connecting map is existentially quantified, as in Stacks 08GW.
Rj_{Y,*}is the chosen total right derived functor of the direct image alongj_Y, and the derived categories are the standard modelHasDerivedCategory.standard, in which Stacks 08GW is stated. The hypotheses are the source's:hWqc,hWqssay thatXis quasi-compact and quasi-separated,hUqc,hUqsthatUis, andhVqc,hVqsthatVis (quasi-separated in the absolute sense, overSpec ℤ, as in Stacks 0CR4).
theorem coherator_edsMayerVietoris_triangle
(hWqc : IsQuasiCompactSpace B S hS W.1)
(hWqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal W.1)
(hUqc : IsQuasiCompactSpace B S hS Uo.toFunctor)
(hUqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal Uo.toFunctor)
(hVqc : IsQuasiCompactSpace B S hS V)
(hVqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal V) :
((qcohDerivedCategoryB B S hS W).ι.IsLeftAdjoint ∧
(qcohDerivedCategoryB B S hS (edsOpen B S hS hsq)).ι.IsLeftAdjoint ∧
(qcohDerivedCategoryB B S hS (edsEtale B S hS hsq)).ι.IsLeftAdjoint ∧
(qcohDerivedCategoryB B S hS (edsInter B S hS hsq)).ι.IsLeftAdjoint) ∧
∀ (DQX : DerivedCategory (Modules.{u, u} B S hS W.1 W.2) ⥤
(qcohDerivedCategoryB B S hS W).FullSubcategory)
(adjX : (qcohDerivedCategoryB B S hS W).ι ⊣ DQX)
(DQU : DerivedCategory (Modules.{u, u} B S hS (edsOpen B S hS hsq).1
(edsOpen B S hS hsq).2) ⥤
(qcohDerivedCategoryB B S hS (edsOpen B S hS hsq)).FullSubcategory)
(adjU : (qcohDerivedCategoryB B S hS (edsOpen B S hS hsq)).ι ⊣ DQU)
(DQV : DerivedCategory (Modules.{u, u} B S hS (edsEtale B S hS hsq).1
(edsEtale B S hS hsq).2) ⥤
(qcohDerivedCategoryB B S hS (edsEtale B S hS hsq)).FullSubcategory)
(adjV : (qcohDerivedCategoryB B S hS (edsEtale B S hS hsq)).ι ⊣ DQV)
(DQP : DerivedCategory (Modules.{u, u} B S hS (edsInter B S hS hsq).1
(edsInter B S hS hsq).2) ⥤
(qcohDerivedCategoryB B S hS (edsInter B S hS hsq)).FullSubcategory)
(adjP : (qcohDerivedCategoryB B S hS (edsInter B S hS hsq)).ι ⊣ DQP)
(K : DerivedCategory (Modules.{u, u} B S hS W.1 W.2)),
∃ δ, Triangle.mk
(biprod.lift
((qcohDerivedCategoryB B S hS W).ι.map (DQX.map
((derivedUnitOfNatTrans (edsAdjOpen B S hS hsq).unit
(edsDerivedRestrictPushOpen B S hS hsq)
(edsTotalRightDerivedUnitOpen B S hS hsq)).app K)) ≫
(coheratorRestrictPushObjIso B S hS (edsOpenHom B S hS hsq)
(isEtale_edsOpenHom B S hS hsq)
(isQuasiCompact_of_isQuasiCompactSpace B S hS B.contains_terminal W.2 hWqs hUqc _)
(isQuasiSeparated_of_isAbsolutelyQuasiSeparated_source B S hS B.contains_terminal
(edsOpen B S hS hsq).2 hUqs _)
(edsDerivedRestrictPushOpen B S hS hsq) (edsTotalRightDerivedUnitOpen B S hS hsq)
DQX adjX DQU adjU K).hom)
((qcohDerivedCategoryB B S hS W).ι.map (DQX.map
((derivedUnitOfNatTrans (edsAdjEtale B S hS hsq).unit
(edsDerivedRestrictPushEtale B S hS hsq)
(edsTotalRightDerivedUnitEtale B S hS hsq)).app K)) ≫
(coheratorRestrictPushObjIso B S hS (edsEtaleHom B S hS hsq)
(isEtale_edsEtaleHom B S hS hsq)
(isQuasiCompact_of_isQuasiCompactSpace B S hS B.contains_terminal W.2 hWqs hVqc _)
(isQuasiSeparated_of_isAbsolutelyQuasiSeparated_source B S hS B.contains_terminal
(edsEtale B S hS hsq).2 hVqs _)
(edsDerivedRestrictPushEtale B S hS hsq) (edsTotalRightDerivedUnitEtale B S hS hsq)
DQX adjX DQV adjV K).hom))
(biprod.desc
((coheratorRestrictPushObjIso B S hS (edsOpenHom B S hS hsq)
(isEtale_edsOpenHom B S hS hsq)
(isQuasiCompact_of_isQuasiCompactSpace B S hS B.contains_terminal W.2 hWqs hUqc _)
(isQuasiSeparated_of_isAbsolutelyQuasiSeparated_source B S hS B.contains_terminal
(edsOpen B S hS hsq).2 hUqs _)
(edsDerivedRestrictPushOpen B S hS hsq) (edsTotalRightDerivedUnitOpen B S hS hsq)
DQX adjX DQU adjU K).inv ≫
(qcohDerivedCategoryB B S hS W).ι.map (DQX.map
((Functor.rightDerivedNatTrans (edsDerivedRestrictPushOpen B S hS hsq)
(edsDerivedRestrictPushInter B S hS hsq)
(edsTotalRightDerivedUnitOpen B S hS hsq)
(edsTotalRightDerivedUnitInter B S hS hsq)
(HomotopyCategory.quasiIso (Modules.{u, u} B S hS W.1 W.2) (ComplexShape.up ℤ))
(Functor.whiskerRight (NatTrans.mapHomotopyCategory
(conjugateEquiv ((edsAdjOpen B S hS hsq).comp
(edsLowerShriekAdjOpen B S hS hsq))
((edsAdjInter B S hS hsq).comp (edsLowerShriekAdjInter B S hS hsq))
(edsShriekOpenComparison B S hS hsq (edsLowerShriekAdjOpen B S hS hsq)
(edsLowerShriekAdjInter B S hS hsq)))
(ComplexShape.up ℤ)) DerivedCategory.Qh)).app K)) ≫
(coheratorRestrictPushObjIso B S hS (edsInterHom B S hS hsq)
(isEtale_edsInterHom B S hS hsq)
(isQuasiCompact_edsInterHom B S hS hsq hWqs hUqc hVqc)
(isQuasiSeparated_edsInterHom B S hS hsq hUqs hVqs)
(edsDerivedRestrictPushInter B S hS hsq) (edsTotalRightDerivedUnitInter B S hS hsq)
DQX adjX DQP adjP K).hom)
((coheratorRestrictPushObjIso B S hS (edsEtaleHom B S hS hsq)
(isEtale_edsEtaleHom B S hS hsq)
(isQuasiCompact_of_isQuasiCompactSpace B S hS B.contains_terminal W.2 hWqs hVqc _)
(isQuasiSeparated_of_isAbsolutelyQuasiSeparated_source B S hS B.contains_terminal
(edsEtale B S hS hsq).2 hVqs _)
(edsDerivedRestrictPushEtale B S hS hsq) (edsTotalRightDerivedUnitEtale B S hS hsq)
DQX adjX DQV adjV K).inv ≫
(qcohDerivedCategoryB B S hS W).ι.map (DQX.map
(-((Functor.rightDerivedNatTrans (edsDerivedRestrictPushEtale B S hS hsq)
(edsDerivedRestrictPushInter B S hS hsq)
(edsTotalRightDerivedUnitEtale B S hS hsq)
(edsTotalRightDerivedUnitInter B S hS hsq)
(HomotopyCategory.quasiIso (Modules.{u, u} B S hS W.1 W.2) (ComplexShape.up ℤ))
(Functor.whiskerRight (NatTrans.mapHomotopyCategory
(conjugateEquiv ((edsAdjEtale B S hS hsq).comp
(edsLowerShriekAdjEtale B S hS hsq))
((edsAdjInter B S hS hsq).comp (edsLowerShriekAdjInter B S hS hsq))
(edsShriekEtaleComparison B S hS hsq (edsLowerShriekAdjEtale B S hS hsq)
(edsLowerShriekAdjInter B S hS hsq)))
(ComplexShape.up ℤ)) DerivedCategory.Qh)).app K))) ≫
(coheratorRestrictPushObjIso B S hS (edsInterHom B S hS hsq)
(isEtale_edsInterHom B S hS hsq)
(isQuasiCompact_edsInterHom B S hS hsq hWqs hUqc hVqc)
(isQuasiSeparated_edsInterHom B S hS hsq hUqs hVqs)
(edsDerivedRestrictPushInter B S hS hsq) (edsTotalRightDerivedUnitInter B S hS hsq)
DQX adjX DQP adjP K).hom))
δ ∈ distTriang (DerivedCategory (Modules.{u, u} B S hS W.1 W.2))
Tag 0CSS, a lemma
Tag 0CSS in the Stacks Project.
AlgebraicGeometry.Space.exists_isZero_homology_coherator_obj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoheratorBounded.lean, line 757.
Stacks 0CSS. Let
Sbe a scheme andXa quasi-compact and quasi-separated algebraic space overS, and letDQ_Xbe the right adjoint of the inclusionD_QCoh(𝒪_X) ⥤ D(𝒪_X)(Stacks 0CR4; any right adjoint, with its adjunction). There is an integerN = N(X)such that for all integersa,band everyKinD(𝒪_X)withH^i(U, K) = 0for every affineUétale overXand everyi ∉ [a, b], the cohomology sheavesH^i(DQ_X(K))vanish fori ∉ [a, b + N].The affine
Uétale overXare the étale chartsUofXwhose scheme is affine, andH^i(U, K)is thei-th cohomology of the complex of sections overUof a K-injective complexIrepresentingK, i.e. of the unbounded derived sectionsRΓ(U, K); the hypothesis is asked of every such representative.
theorem exists_isZero_homology_coherator_obj (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(DQ : DerivedCategory (Modules.{u, u} B S hS X.1 X.2) ⥤
(DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).FullSubcategory)
(adj : (DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι ⊣ DQ) :
∃ N : ℤ, ∀ (a b : ℤ) (K : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)),
(∀ U : SmallEtale B S hS X.1, IsAffineChart B S hS X.1 U →
∀ I : CochainComplex (Modules.{u, u} B S hS X.1 X.2) ℤ, I.IsKInjective →
Nonempty (DerivedCategory.Q.obj I ≅ K) → ∀ i : ℤ, i < a ∨ b < i →
IsZero ((((chartSectionsAb B S hS X U).mapHomologicalComplex
(ComplexShape.up ℤ)).obj I).homology i)) →
∀ i : ℤ, i < a ∨ b + N < i →
IsZero ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) i).obj
((DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι.obj (DQ.obj K)))
Tag 0CST, an example
Tag 0CST in the Stacks Project.
AlgebraicGeometry.Space.isDerivedLimit_coherator_obj, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoheratorInverseLimit.lean, line 107. The attribute's comment: “assertion DQ_X(R lim F_n) = (R lim in D_QCoh(O_X))(F_n): for every right adjoint DQ of the inclusion D_QCoh(O_X) -> D(O_X), at every model of D(O_X), and every inverse system in D_QCoh(O_X); DQ_X exists on a qcqs X by 0CR4”.
Stacks 0CST, the coherator commutes with derived limits. Let
Xbe an algebraic space overS,DQa right adjoint of the inclusionι : D_QCoh(𝒪_X) ⥤ D(𝒪_X)andFan inverse system inD_QCoh(𝒪_X). IfLis a derived limit ofFcomputed inD(𝒪_X), then the product ofFexists inD_QCoh(𝒪_X)andDQ Lis a derived limit ofFthere:DQ_X(R lim Fₙ) = (R lim in D_QCoh(𝒪_X))(Fₙ).Such a
DQexists whenXis quasi-compact and quasi-separated, by Stacks 0CR4 (AlgebraicGeometry.Space.isLeftAdjoint_ι_derivedCategoryB_isQuasicoherentOnSite); the system of the source is the system of quasi-coherent sheaves(𝓕ₙ), placed in degree0.
theorem isDerivedLimit_coherator_obj
(DQ : DerivedCategory (Modules.{u, u} B S hS X.1 X.2) ⥤
(DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).FullSubcategory)
(adj : (DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι ⊣ DQ)
(F : ℕᵒᵖ ⥤ (DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).FullSubcategory)
[HasProduct fun n : ℕ ↦ (F ⋙ (DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι).obj (op n)]
(L : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hL : IsDerivedLimit (F ⋙ (DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι) L) :
∃ h : HasProduct fun n : ℕ ↦ F.obj (op n),
letI := h
IsDerivedLimit F (DQ.obj L)
AlgebraicGeometry.Space.nonempty_homology_chartSections_iso_of_isDerivedLimit_single, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoheratorInverseLimit.lean, line 254. The attribute's comment: “assertion H^i(U, K) = H^i(R lim Gamma(U, F_n)) for K = R lim F_n in D(O_X) and U affine etale over X; R lim Gamma(U, F_n) is any derived limit in D(Ab), at any models; H^i(U, K) is computed on a K-injective representative, as in 0CSS”.
Stacks 0CST,
H^i(U, K) = H^i(R lim Γ(U, 𝓕ₙ)). Let(𝓕ₙ)be an inverse system of quasi-coherent𝒪_X-modules on an algebraic spaceX, letL = R lim 𝓕ₙbe a derived limit inD(𝒪_X)of the system(𝓕ₙ)placed in degree0, and letUbe an affine étale chart ofX. Then for every derived limitR = R lim Γ(U, 𝓕ₙ)inD(Ab)of the abelian groupsΓ(U, 𝓕ₙ)placed in degree0, and everyi,H^i(U, L) ≅ H^i(R).
H^i(U, L)is the cohomology of the complex of sections overUof any K-injective complexIrepresentingL.
theorem nonempty_homology_chartSections_iso_of_isDerivedLimit_single
(𝓕 : ℕᵒᵖ ⥤ Modules.{u, u} B S hS X.1 X.2)
(hqc : ∀ n : ℕ, (𝓕.obj (op n)).IsQuasicoherentOnSite)
[HasProduct fun n : ℕ ↦
(𝓕 ⋙ DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).obj (op n)]
(L : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hL : IsDerivedLimit (𝓕 ⋙ DerivedCategory.singleFunctor _ 0) L)
(U : SmallEtale B S hS X.1) (hU : IsAffineChart B S hS X.1 U)
(I : CochainComplex (Modules.{u, u} B S hS X.1 X.2) ℤ) (hI : I.IsKInjective)
(eI : Nonempty (DerivedCategory.Q.obj I ≅ L))
[HasProduct fun n : ℕ ↦ (𝓕 ⋙ chartSectionsAb B S hS X U ⋙
DerivedCategory.singleFunctor AddCommGrpCat.{u} 0).obj (op n)]
(R : DerivedCategory AddCommGrpCat.{u})
(hR : IsDerivedLimit (𝓕 ⋙ chartSectionsAb B S hS X U ⋙
DerivedCategory.singleFunctor AddCommGrpCat.{u} 0) R)
(i : ℤ) :
Nonempty ((((chartSectionsAb B S hS X U).mapHomologicalComplex (ComplexShape.up ℤ)).obj
I).homology i ≅ (DerivedCategory.homologyFunctor AddCommGrpCat.{u} i).obj R)
AlgebraicGeometry.Space.isZero_homology_chartSections_of_isDerivedLimit_single, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoheratorInverseLimit.lean, line 289. The attribute's comment: “assertion H^i(U, R lim F_n) = 0 for i outside [0, 1] and U affine etale over X, at any model of D(O_X)”.
Stacks 0CST,
H^i(U, K) = 0unlessi ∈ [0, 1]. Let(𝓕ₙ)be an inverse system of quasi-coherent𝒪_X-modules on an algebraic spaceXandL = R lim 𝓕ₙa derived limit inD(𝒪_X)of the system(𝓕ₙ)placed in degree0. ThenH^i(U, L) = 0for every affine étale chartUofXand everyi ∉ [0, 1].
theorem isZero_homology_chartSections_of_isDerivedLimit_single
(𝓕 : ℕᵒᵖ ⥤ Modules.{u, u} B S hS X.1 X.2)
(hqc : ∀ n : ℕ, (𝓕.obj (op n)).IsQuasicoherentOnSite)
[HasProduct fun n : ℕ ↦
(𝓕 ⋙ DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).obj (op n)]
(L : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hL : IsDerivedLimit (𝓕 ⋙ DerivedCategory.singleFunctor _ 0) L)
(U : SmallEtale B S hS X.1) (hU : IsAffineChart B S hS X.1 U)
(I : CochainComplex (Modules.{u, u} B S hS X.1 X.2) ℤ) (hI : I.IsKInjective)
(eI : Nonempty (DerivedCategory.Q.obj I ≅ L)) (i : ℤ) (hi : i < 0 ∨ 1 < i) :
IsZero ((((chartSectionsAb B S hS X U).mapHomologicalComplex (ComplexShape.up ℤ)).obj
I).homology i)
AlgebraicGeometry.Space.exists_isZero_homology_coherator_obj_of_isDerivedLimit_single, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedCoheratorInverseLimit.lean, line 344. The attribute's comment: “conclusion: the R lim in D_QCoh(O_X), i.e. DQ_X(R lim F_n), lies in D^b_QCoh(O_X) and has vanishing cohomology sheaves in negative degrees (in [0, 1 + N], N = N(X) uniform), by 0CSS; at the standard model, the one of 0CSS”.
Stacks 0CST,
DQ_X(R lim 𝓕ₙ)is inD^b_QCoh(𝒪_X)with no negative cohomology. LetXbe a quasi-compact and quasi-separated algebraic space overSandDQa right adjoint of the inclusionD_QCoh(𝒪_X) ⥤ D(𝒪_X). There is an integerNsuch that for every inverse system(𝓕ₙ)of quasi-coherent𝒪_X-modules and every derived limitL = R lim 𝓕ₙinD(𝒪_X)of(𝓕ₙ)placed in degree0, the cohomology sheavesH^i(DQ L)vanish fori < 0and fori > 1 + N. ByAlgebraicGeometry.Space.isDerivedLimit_coherator_obj,DQ Lis theR limof(𝓕ₙ)inD_QCoh(𝒪_X).This is Stacks 0CSS (
AlgebraicGeometry.Space.exists_isZero_homology_coherator_obj) with[a, b] = [0, 1].
theorem exists_isZero_homology_coherator_obj_of_isDerivedLimit_single (X : AlgebraicSpaces B S hS)
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(DQ : DerivedCategory (Modules.{u, u} B S hS X.1 X.2) ⥤
(DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).FullSubcategory)
(adj : (DerivedCategoryB.derivedCategoryB (SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι ⊣ DQ) :
∃ N : ℤ, ∀ (𝓕 : ℕᵒᵖ ⥤ Modules.{u, u} B S hS X.1 X.2),
(∀ n : ℕ, (𝓕.obj (op n)).IsQuasicoherentOnSite) →
∀ [HasProduct fun n : ℕ ↦
(𝓕 ⋙ DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).obj (op n)]
(L : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)),
IsDerivedLimit (𝓕 ⋙ DerivedCategory.singleFunctor _ 0) L →
∀ i : ℤ, i < 0 ∨ 1 + N < i →
IsZero ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) i).obj
((DerivedCategoryB.derivedCategoryB
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
(ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))).ι.obj (DQ.obj L)))
Sources
Section75.20 Cohomology and base change, IV#
Section 75.20 of the Stacks Project, Tag 08IM. 7 tags here are carried by 14 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 08IN, a lemma
Tag 08IN in the Stacks Project.
AlgebraicGeometry.Space.isIso_smallEtaleProjectionFormulaMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/SmallEtaleProjectionFormulaIso.lean, line 126.
Stacks 08IN. Let
f : X ⟶ Ybe a quasi-compact and quasi-separated morphism of algebraic spaces,E ∈ D_QCoh(𝒪_X)andK ∈ D_QCoh(𝒪_Y). Then the projection formula mapRf_*(E) ⊗^L K ⟶ Rf_*(E ⊗^L Lf^*K)is an isomorphism.
theorem isIso_smallEtaleProjectionFormulaMap (hfqc : IsQuasiCompact B S hS f.hom)
(hfqs : IsQuasiSeparated B S hS f.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : QcohDerived B S hS X E)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)))
(hK : QcohDerived B S hS Y K) :
IsIso (smallEtaleProjectionFormulaMap B S hS f E K)
Tag 08IP, a definition
Tag 08IP in the Stacks Project.
AlgebraicGeometry.Space.TorIndependent, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependence.lean, line 58.
Stacks 08IP. Two algebraic spaces are Tor independent over a base when the geometric étale local rings at every compatible pair of points are Tor independent for the homomorphisms induced on germs by their structure morphisms.
public def TorIndependent {B : BigFppfSite.{u}} {S : Scheme.{u}}
{hS : B.Contains S} {F G H : (BigFppfOver B S hS)ᵒᵖ ⥤ Type u}
(_hF : IsAlgebraicSpace B S hS F) (_hG : IsAlgebraicSpace B S hS G)
(_hH : IsAlgebraicSpace B S hS H)
(f : F ⟶ H) (g : G ⟶ H) : Prop
Tag 08IQ, a lemma
Tag 08IQ in the Stacks Project.
AlgebraicGeometry.Space.torIndependent_iff_forall_etale_diagram, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependenceEtaleLocal.lean, line 382. The attribute's comment: “(1) <=> (2)”.
Stacks 08IQ, (1) ⟺ (2). Let
XandYbe algebraic spaces over an algebraic spaceB. ThenXandYare Tor independent overBif and only if for every commutative diagramwith étale vertical arrows,U ⟶ W ⟵ V ↓ ↓ ↓ X ⟶ B ⟵ YUandVare Tor independent overW.Condition (3), which asks for a single such diagram in which
W ⟶ Bis surjective and the two upper terms are étale surjective over the base changesX ×_B WandY ×_B W, is equivalent to condition (1): it implies it byAlgebraicGeometry.Space.torIndependent_of_isEtale_surjective_diagramand follows from it byAlgebraicGeometry.Space.exists_isEtale_surjective_diagram_of_torIndependent. Condition (4), which reads the conclusion forU,VandWschemes in the sense of Stacks 08IA, that is on ordinary local rings, implies condition (1) byAlgebraicGeometry.Space.torIndependent_of_torIndependent_scheme_diagraminStacksFrontier.AlgebraicGeometry.Spaces.TorIndependenceSchemeComparison, and follows from condition (1) byAlgebraicGeometry.Space.exists_torIndependent_scheme_diagram_of_torIndependentinStacksFrontier.AlgebraicGeometry.Spaces.TorIndependenceSchemeConverse. So the four conditions of Stacks 08IQ are equivalent.
public theorem torIndependent_iff_forall_etale_diagram {Bs X Y : AlgebraicSpaces B S hS}
(fX : X ⟶ Bs) (gY : Y ⟶ Bs) :
TorIndependent X.2 Y.2 Bs.2 fX.hom gY.hom ↔
∀ (U V W : AlgebraicSpaces B S hS) (a : U ⟶ X) (b : V ⟶ Y) (c : W ⟶ Bs)
(uw : U ⟶ W) (vw : V ⟶ W), IsEtale B S hS a.hom → IsEtale B S hS b.hom →
IsEtale B S hS c.hom → uw.hom ≫ c.hom = a.hom ≫ fX.hom →
vw.hom ≫ c.hom = b.hom ≫ gY.hom → TorIndependent U.2 V.2 W.2 uw.hom vw.hom
AlgebraicGeometry.Space.torIndependent_of_isEtale_surjective_diagram, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependenceEtaleLocal.lean, line 524. The attribute's comment: “(3) => (1)”.
Stacks 08IQ, (3) ⟹ (1). Let
XandYbe algebraic spaces over an algebraic spaceBs. Suppose given one commutative diagramwith étale vertical arrows in whichU ⟶ W ⟵ V ↓ ↓ ↓ X ⟶ Bs ⟵ YW ⟶ Bsis surjective and in whichUandVare étale and surjective over the fibre productsX ×_{Bs} WandY ×_{Bs} W. IfUandVare Tor independent overW, thenXandYare Tor independent overBs.The two fibre products are presented as arbitrary cartesian squares of the underlying presheaves, which is how a fibre product of algebraic spaces is formed.
public theorem torIndependent_of_isEtale_surjective_diagram
{Bs X Y U V W PX PY : AlgebraicSpaces B S hS}
(fX : X ⟶ Bs) (gY : Y ⟶ Bs) (c : W ⟶ Bs)
(a : U ⟶ X) (uw : U ⟶ W) (b : V ⟶ Y) (vw : V ⟶ W)
(pX : PX ⟶ X) (pW : PX ⟶ W) (qY : PY ⟶ Y) (qW : PY ⟶ W)
(u' : U ⟶ PX) (v' : V ⟶ PY)
(sqX : IsPullback pX.hom pW.hom fX.hom c.hom)
(sqY : IsPullback qY.hom qW.hom gY.hom c.hom)
(hu'X : u' ≫ pX = a) (hu'W : u' ≫ pW = uw)
(hv'Y : v' ≫ qY = b) (hv'W : v' ≫ qW = vw)
(hcEtale : IsEtale B S hS c.hom) (hcSurj : IsSurjective B S hS c.hom)
(haEtale : IsEtale B S hS a.hom) (hbEtale : IsEtale B S hS b.hom)
(huEtale : IsEtale B S hS u'.hom) (huSurj : IsSurjective B S hS u'.hom)
(hvEtale : IsEtale B S hS v'.hom) (hvSurj : IsSurjective B S hS v'.hom)
(h : TorIndependent U.2 V.2 W.2 uw.hom vw.hom) :
TorIndependent X.2 Y.2 Bs.2 fX.hom gY.hom
AlgebraicGeometry.Space.exists_isEtale_surjective_diagram_of_torIndependent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependenceEtaleLocal.lean, line 655. The attribute's comment: “(1) => (3)”.
Stacks 08IQ, (1) ⟹ (3). Let
XandYbe algebraic spaces over an algebraic spaceBswhich are Tor independent overBs. Then there is a commutative diagramwith étale vertical arrows in whichU ⟶ W ⟵ V ↓ ↓ ↓ X ⟶ Bs ⟵ YW ⟶ Bsis surjective, in whichUandVare étale and surjective over the fibre productsX ×_{Bs} WandY ×_{Bs} W, and in whichUandVare Tor independent overW: this is the existence asserted by condition (3).The witness is the diagram of identities,
W = Bs,U = XandV = Y, whose two fibre products areXandYthemselves byCategoryTheory.IsPullback.of_id_fst; the Tor independence asked of the diagram is then the hypothesis on the nose. The converse implication isAlgebraicGeometry.Space.torIndependent_of_isEtale_surjective_diagram.
public theorem exists_isEtale_surjective_diagram_of_torIndependent
{Bs X Y : AlgebraicSpaces B S hS} (fX : X ⟶ Bs) (gY : Y ⟶ Bs)
(h : TorIndependent X.2 Y.2 Bs.2 fX.hom gY.hom) :
∃ (U V W PX PY : AlgebraicSpaces B S hS) (c : W ⟶ Bs) (a : U ⟶ X) (uw : U ⟶ W)
(b : V ⟶ Y) (vw : V ⟶ W) (pX : PX ⟶ X) (pW : PX ⟶ W) (qY : PY ⟶ Y) (qW : PY ⟶ W)
(u' : U ⟶ PX) (v' : V ⟶ PY),
IsPullback pX.hom pW.hom fX.hom c.hom ∧ IsPullback qY.hom qW.hom gY.hom c.hom ∧
u' ≫ pX = a ∧ u' ≫ pW = uw ∧ v' ≫ qY = b ∧ v' ≫ qW = vw ∧
IsEtale B S hS c.hom ∧ IsSurjective B S hS c.hom ∧
IsEtale B S hS a.hom ∧ IsEtale B S hS b.hom ∧
IsEtale B S hS u'.hom ∧ IsSurjective B S hS u'.hom ∧
IsEtale B S hS v'.hom ∧ IsSurjective B S hS v'.hom ∧
TorIndependent U.2 V.2 W.2 uw.hom vw.hom
AlgebraicGeometry.Space.torIndependent_of_torIndependent_scheme_diagram, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependenceSchemeComparison.lean, line 595. The attribute's comment: “(4) => (1)”.
Stacks 08IQ, (4) ⟹ (1). Let
XandYbe algebraic spaces over an algebraic spaceBs. Suppose given one commutative diagramin whichU ⟶ W ⟵ V ↓ ↓ ↓ X ⟶ Bs ⟵ YU,VandWare schemes, the vertical arrows are étale,W ⟶ Bsis surjective andUandVare étale and surjective over the fibre productsX ×_{Bs} WandY ×_{Bs} W. IfUandVare Tor independent overWin the sense of Stacks 08IA, that is on their ordinary local rings, thenXandYare Tor independent overBs.This is condition (4) of Stacks 08IQ: a diagram as in condition (3) whose three upper terms are schemes, with Tor independence read in the sense of Derived Categories of Schemes, Definition 08IA. The passage from the ordinary local rings of the three schemes to the étale local rings of the algebraic spaces they represent is
AlgebraicGeometry.Space.torIndependent_yoneda_of_torIndependent_scheme; condition (3) then gives condition (1) byAlgebraicGeometry.Space.torIndependent_of_isEtale_surjective_diagram.
theorem torIndependent_of_torIndependent_scheme_diagram
{Bs X Y PX PY : AlgebraicSpaces B S hS} (fX : X ⟶ Bs) (gY : Y ⟶ Bs)
{U₀ V₀ W₀ : BigFppfOver B S hS}
(c : yoneda.obj W₀ ⟶ Bs.1) (a : yoneda.obj U₀ ⟶ X.1) (uw : U₀ ⟶ W₀)
(b : yoneda.obj V₀ ⟶ Y.1) (vw : V₀ ⟶ W₀)
(pX : PX ⟶ X) (pW : PX.1 ⟶ yoneda.obj W₀) (qY : PY ⟶ Y) (qW : PY.1 ⟶ yoneda.obj W₀)
(u' : yoneda.obj U₀ ⟶ PX.1) (v' : yoneda.obj V₀ ⟶ PY.1)
(sqX : IsPullback pX.hom pW fX.hom c) (sqY : IsPullback qY.hom qW gY.hom c)
(hu'X : u' ≫ pX.hom = a) (hu'W : u' ≫ pW = yoneda.map uw)
(hv'Y : v' ≫ qY.hom = b) (hv'W : v' ≫ qW = yoneda.map vw)
(hcEtale : IsEtale B S hS c) (hcSurj : IsSurjective B S hS c)
(haEtale : IsEtale B S hS a) (hbEtale : IsEtale B S hS b)
(huEtale : IsEtale B S hS u') (huSurj : IsSurjective B S hS u')
(hvEtale : IsEtale B S hS v') (hvSurj : IsSurjective B S hS v')
(h : AlgebraicGeometry.TorIndependent ((BigFppfOver.toScheme B S hS).map uw)
((BigFppfOver.toScheme B S hS).map vw)) :
TorIndependent X.2 Y.2 Bs.2 fX.hom gY.hom
AlgebraicGeometry.Space.exists_torIndependent_scheme_diagram_of_torIndependent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/TorIndependenceSchemeConverse.lean, line 601. The attribute's comment: “(1) => (4)”.
Stacks 08IQ, (1) ⟹ (4). Let
XandYbe algebraic spaces over an algebraic spaceBswhich are Tor independent overBs. Then there is a commutative diagramin whichU ⟶ W ⟵ V ↓ ↓ ↓ X ⟶ Bs ⟵ YU,VandWare schemes, the vertical arrows are étale,W ⟶ Bsis surjective,UandVare étale and surjective over the fibre productsX ×_{Bs} WandY ×_{Bs} W, andUandVare Tor independent overWin the sense of Stacks 08IA — that is, on their ordinary local rings. This is the existence asserted by condition (4).The diagram is the atlas tower: a scheme chart
W ⟶ Bsof the base, then scheme charts of the two fibre products, which are algebraic spaces. Condition (2) of Stacks 08IQ gives Tor independence of the étale local rings over that diagram, andAlgebraicGeometry.Space.torIndependent_scheme_of_torIndependent_yonedaturns it into Tor independence of the ordinary local rings. The converse implication isAlgebraicGeometry.Space.torIndependent_of_torIndependent_scheme_diagram.
theorem exists_torIndependent_scheme_diagram_of_torIndependent
{Bs X Y : AlgebraicSpaces B S hS} (fX : X ⟶ Bs) (gY : Y ⟶ Bs)
(h : TorIndependent X.2 Y.2 Bs.2 fX.hom gY.hom) :
∃ (U₀ V₀ W₀ : BigFppfOver B S hS) (PX PY : AlgebraicSpaces B S hS)
(c : yoneda.obj W₀ ⟶ Bs.1) (a : yoneda.obj U₀ ⟶ X.1) (uw : U₀ ⟶ W₀)
(b : yoneda.obj V₀ ⟶ Y.1) (vw : V₀ ⟶ W₀)
(pX : PX ⟶ X) (pW : PX.1 ⟶ yoneda.obj W₀) (qY : PY ⟶ Y) (qW : PY.1 ⟶ yoneda.obj W₀)
(u' : yoneda.obj U₀ ⟶ PX.1) (v' : yoneda.obj V₀ ⟶ PY.1),
IsPullback pX.hom pW fX.hom c ∧ IsPullback qY.hom qW gY.hom c ∧
u' ≫ pX.hom = a ∧ u' ≫ pW = yoneda.map uw ∧ v' ≫ qY.hom = b ∧
v' ≫ qW = yoneda.map vw ∧
IsEtale B S hS c ∧ IsSurjective B S hS c ∧
IsEtale B S hS a ∧ IsEtale B S hS b ∧
IsEtale B S hS u' ∧ IsSurjective B S hS u' ∧
IsEtale B S hS v' ∧ IsSurjective B S hS v' ∧
AlgebraicGeometry.TorIndependent ((BigFppfOver.toScheme B S hS).map uw)
((BigFppfOver.toScheme B S hS).map vw)
Tag 08IR, a lemma
Tag 08IR in the Stacks Project.
AlgebraicGeometry.Space.isIso_smallEtaleDerivedBaseChangeMap_of_torIndependent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/SmallEtaleDerivedTorIndependentBaseChange.lean, line 248.
Stacks 08IR. Let
g : Y' ⟶ Ybe a morphism of algebraic spaces andf : X ⟶ Ya quasi-compact and quasi-separated one, with base changef' : X' ⟶ Y',g' : X' ⟶ X. IfXandY'are Tor independent overY, then the base change mapLg^*Rf_*E ⟶ Rf'_*L(g')^*Eis an isomorphism for everyE ∈ D_QCoh(𝒪_X).
theorem isIso_smallEtaleDerivedBaseChangeMap_of_torIndependent (sq : IsPullback g' f' f g)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hTor : TorIndependent X.2 Y'.2 Y.2 f.hom g.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : QcohDerived B S hS X E) :
IsIso (smallEtaleDerivedModuleBaseChangeMap B S hS f g f' g' sq.w E)
Tag 0E4S, a lemma
Tag 0E4S in the Stacks Project.
AlgebraicGeometry.Space.isIso_globalDerivedHomBaseChangeMap_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GlobalDerivedHomBaseChangeIso.lean, line 93. The attribute's comment: “case (1): M perfect, K in D_QCoh”.
Stacks 0E4S, case (1). Let
g : S' ⟶ Sbe a morphism of affine schemes,R = Γ(S, 𝒪),R' = Γ(S', 𝒪), and letX' = X ×_S S'forf : X ⟶ Squasi-compact and quasi-separated withXandS'Tor independent overS. ForM ∈ D(𝒪_X)perfect andK ∈ D_QCoh(𝒪_X)the canonical mapRHom_X(M, K) ⊗^L_R R' ⟶ RHom_{X'}(L(g')^*M, L(g')^*K)is an isomorphism.
theorem isIso_globalDerivedHomBaseChangeMap_of_isPerfect
(sq : IsPullback g' f' f (representedSpaceMap B S hS g))
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hTor : TorIndependent X.2 (isAlgebraicSpace_yoneda B S hS U')
(isAlgebraicSpace_yoneda B S hS U) f.hom (representedSpaceMap B S hS g).hom)
(M K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hM : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) M)
(hK : QcohDerived B S hS X K) :
IsIso (globalDerivedHomBaseChangeMap B S hS g f f' g' sq.w M K)
AlgebraicGeometry.Space.isIso_globalDerivedHomBaseChangeMap_of_isPseudoCoherent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GlobalDerivedHomBaseChangeIso.lean, line 118. The attribute's comment: “case (2): M pseudo-coherent, K in D^+_QCoh, R' of finite tor dimension over R”.
Stacks 0E4S, case (2). Let
g : S' ⟶ Sbe a morphism of affine schemes,R = Γ(S, 𝒪),R' = Γ(S', 𝒪), and letX' = X ×_S S'forf : X ⟶ Squasi-compact and quasi-separated withXandS'Tor independent overS. IfR'has finite Tor dimension overR, then forM ∈ D(𝒪_X)pseudo-coherent andK ∈ D^+_QCoh(𝒪_X)the canonical mapRHom_X(M, K) ⊗^L_R R' ⟶ RHom_{X'}(L(g')^*M, L(g')^*K)is an isomorphism.
theorem isIso_globalDerivedHomBaseChangeMap_of_isPseudoCoherent
(sq : IsPullback g' f' f (representedSpaceMap B S hS g))
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hTor : TorIndependent X.2 (isAlgebraicSpace_yoneda B S hS U')
(isAlgebraicSpace_yoneda B S hS U) f.hom (representedSpaceMap B S hS g).hom)
(hR' : AlgebraicGeometry.HasFiniteTorDimensionRingHom
(representedAffineCommRingCatHom B S hS g).hom)
(M K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hM : RingedSite.IsPseudoCoherent (smallEtaleRingedSite B S hS X.1 X.2) M)
(hK : DerivedCategoryB.derivedCategoryBPlus
(A := SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
((Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))) K) :
IsIso (globalDerivedHomBaseChangeMap B S hS g f f' g' sq.w M K)
Tag 0E4T, a remark
Tag 0E4T in the Stacks Project.
AlgebraicGeometry.Space.globalDerivedHomBaseChangeMap_comp_map_counit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GlobalDerivedHomBaseChange.lean, line 236.
For
M ∈ D(𝒪_X)andL ∈ D(𝒪_{X'}), the canonical map of Stacks 0E4S at(M, Rg'_*L)followed byRΓ(X', -)ofR𝓗om(L(g')^*M, -)of the counitL(g')^*Rg'_*L ⟶ Lis the multiplication mapK' ⊗^L_R R' ⟶ K', the counit of- ⊗^L_R R' ⊣restriction of scalars, atK' = RHom_{X'}(L(g')^*M, L), after the identificationRHom_X(M, Rg'_*L) = RHom_{X'}(L(g')^*M, L).
theorem globalDerivedHomBaseChangeMap_comp_map_counit
(M : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(L : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X'))) :
globalDerivedHomBaseChangeMap B S hS g f f' g' w M
((derivedPushforward (X := smallEtaleRingedTopos B S hS X')
(Y := smallEtaleRingedTopos B S hS X) (smallEtaleRingedToposHom B S hS g')).obj L) ≫
(representedAffineDerivedSections B S hS U').map ((derivedModulePushforward B S hS f').map
(((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj M))).map
((smallEtaleDerivedModuleAdjunction B S hS g').counit.app L))) =
(representedAffineDerivedExtension B S hS g).functor.map
((derivedModulePushforward B S hS f ⋙ representedAffineDerivedSections B S hS U).map
((derivedModulePushforwardSheafHomIso g' M).inv.app L) ≫
(affineBaseDerivedSectionsPushforwardIso B S hS g f f' g' w).hom.app _) ≫
(RingTheory.TorIndependence.totalDerivedExtensionScalarsAdjunction
(representedAffineCommRingCatHom B S hS g).hom).counit.app _
Tag 0DKD, a lemma
Tag 0DKD in the Stacks Project.
AlgebraicGeometry.Space.hasTorAmplitude_derivedScalarRestriction_smallEtaleDerivedModulePullback, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/TorIndependentDerivedPullback.lean, line 608. The attribute's comment: “part (1)”.
Stacks 0DKD (1). Let
X' → X,Y' → Ybe a cartesian square of algebraic spaces overSwithXandY'Tor independent overY. IfE ∈ D(𝒪_X)has Tor amplitude in[a, b]as a complex off⁻¹𝒪_Y-modules, thenL(g')^*Ehas Tor amplitude in[a, b]as a complex off'⁻¹𝒪_{Y'}-modules.
theorem hasTorAmplitude_derivedScalarRestriction_smallEtaleDerivedModulePullback
(htor : TorIndependent X.2 Y'.2 Y.2 f.hom g.hom)
(T : Stacks.SheafOfModules.DerivedTensorProduct (inverseImageRingedSite B S hS f))
(T' : Stacks.SheafOfModules.DerivedTensorProduct (inverseImageRingedSite B S hS f'))
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2)) (a b : ℤ)
(hE : HasTorAmplitude T ((derivedScalarRestriction B S hS f).obj E) a b) :
HasTorAmplitude T' ((derivedScalarRestriction B S hS f').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)) a b
AlgebraicGeometry.Space.isIso_smallEtaleDerivedModulePullbackComparison_of_isFlatOver, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/TorIndependentDerivedPullback.lean, line 764. The attribute's comment: “part (2)”.
Stacks 0DKD (2). For a cartesian square of algebraic spaces with
XandY'Tor independent overY, and a quasi-coherent𝒪_X-module𝒢flat overY(Stacks 05VV), the canonical comparisonL(g')^*𝒢 ⟶ (g')^*𝒢is an isomorphism.
theorem isIso_smallEtaleDerivedModulePullbackComparison_of_isFlatOver
(htor : TorIndependent X.2 Y'.2 Y.2 f.hom g.hom) (G : Modules.{u, u} B S hS X.1 X.2)
[G.IsQuasicoherentOnSite] (hG : IsFlatOver B S hS X.2 f.hom G) :
IsIso ((smallEtaleDerivedModulePullbackComparison B S hS g').app
((HomotopyCategory.quotient _ (ComplexShape.up ℤ)).obj
((CochainComplex.singleFunctor
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) 0).obj G)))
AlgebraicGeometry.Space.isIso_smallEtaleDerivedModulePullbackComparison_of_ringedToposHom_isFlatOver, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/TorIndependentDerivedPullbackRingedFlat.lean, line 149. The attribute's comment: “part (2)”.
For a cartesian square of algebraic spaces with
XandY'Tor independent overY, and an𝒪_X-module𝒢flat overY— flat as anf⁻¹𝒪_Y-module, Stacks 08M5 — the canonical comparisonL(g')^*𝒢 ⟶ (g')^*𝒢is an isomorphism.
theorem isIso_smallEtaleDerivedModulePullbackComparison_of_ringedToposHom_isFlatOver
(htor : TorIndependent X.2 Y'.2 Y.2 f.hom g.hom)
(G : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(hG : RingedToposHom.IsFlatOver (X := smallEtaleRingedTopos B S hS X)
(Y := smallEtaleRingedTopos B S hS Y) (smallEtaleRingedToposHom B S hS f) G) :
IsIso ((smallEtaleDerivedModulePullbackComparison B S hS g').app
((HomotopyCategory.quotient _ (ComplexShape.up ℤ)).obj
((CochainComplex.singleFunctor
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) 0).obj G)))
Sources
Section75.21 Cohomology and base change, V#
Section 75.21 of the Stacks Project, Tag 0DKE. 6 tags here are carried by 10 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0DKF, an equation
Tag 0DKF in the Stacks Project.
AlgebraicGeometry.Space.geometricStalkBaseChangeCohomologyMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/GeometricStalkCohomologyMap.lean, line 280. The attribute's comment: “the map, given the derived pullback-stalk comparison ps”.
Stacks Project, Equation 0DKF. The canonical map
H^n(K_x̄ ⊗ᴸ_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'}) ⊗_{(𝒪_{X,x̄} ⊗_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'})} 𝒪_{X',x̄'} ⟶ H^n(K'_x̄')of
𝒪_{X',x̄'}-modules attached to a commutative square of algebraic spaces, a geometric pointx̄'ofX', objectsKofD(𝒪_X)andK'ofD(𝒪_{X'})and a morphismφ : L(g')^*K ⟶ K'. No cartesianness, Tor independence or quasi-coherence is required.It is built, not chosen: the
𝒪_{X,x̄} ⊗_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'}-module structure on the source isRingTheory.TorIndependence.baseChangeCohomologyModuleand the map isRingTheory.TorIndependence.ringBaseChangeCohomologyMap, the adjoint of the Stacks 065Z comparison followed byφ.Two premises are carried, and the construction is complete relative to them. The instance argument
Functor.HasLeftDerivedFunctoris the existence ofL(g')^*itself, carried asAlgebraicGeometry.Space.smallEtaleDerivedModulePullbackcarries it, because the K-flat resolutions of Stacks 06YS are available in this tree only on a site which isCategoryTheory.SmallCategory. The morphismpsis the derived pullback-stalk comparisonK_x̄ ⊗ᴸ_{𝒪_{X,x̄}} 𝒪_{X',x̄'} ⟶ (L(g')^*K)_x̄', which the source states as an equality and which nothing in this tree constructs for algebraic spaces.
def geometricStalkBaseChangeCohomologyMap
(f : X ⟶ Y) (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X) (w : g' ≫ f = f' ≫ g)
(x' : GeometricPoint B S hS X'.1)
[(smallEtaleModulePullbackOnHomotopy B S hS g').HasLeftDerivedFunctor
(smallEtaleModuleQuasiIso B S hS X)]
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(ps : (CochainComplex.DerivedExtensionScalars.totalDerivedExtensionScalars.{u, u, u}
(stalkStructureMap B S hS g' x').hom).functor.obj
((geometricDerivedStalk B S hS (geometricPointMap g'.hom x')).obj K) ⟶
(geometricDerivedStalk B S hS x').obj
((smallEtaleDerivedModulePullback B S hS g').obj K))
(n : ℤ)
AlgebraicGeometry.Space.canonicalGeometricStalkBaseChangeCohomologyMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/GeometricStalkCohomologyMapCanonical.lean, line 65.
Stacks Project, Equation 0DKF. The canonical map
H^n(K_x̄ ⊗ᴸ_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'}) ⊗_{(𝒪_{X,x̄} ⊗_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'})} 𝒪_{X',x̄'} ⟶ H^n(K'_x̄')of
𝒪_{X',x̄'}-modules attached to a commutative square of algebraic spaces, a geometric pointx̄'ofX', objectsKofD(𝒪_X)andK'ofD(𝒪_{X'})and a morphismφ : L(g')^*K ⟶ K'. No cartesianness, Tor independence or quasi-coherence is required.It is built, not chosen: the
𝒪_{X,x̄} ⊗_{𝒪_{Y,ȳ}} 𝒪_{Y',ȳ'}-module structure on the source isRingTheory.TorIndependence.baseChangeCohomologyModuleand the map isRingTheory.TorIndependence.ringBaseChangeCohomologyMap, the adjoint of the Stacks 065Z comparison, precomposed with the derived pullback-stalk comparisonK_x̄ ⊗ᴸ_{𝒪_{X,x̄}} 𝒪_{X',x̄'} ⟶ (L(g')^*K)_x̄'and followed byφ.
def canonicalGeometricStalkBaseChangeCohomologyMap
(f : X ⟶ Y) (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X) (w : g' ≫ f = f' ≫ g)
(x' : GeometricPoint B S hS X'.1)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(n : ℤ)
Tag 0DKG, a lemma
Tag 0DKG in the Stacks Project.
AlgebraicGeometry.Space.tfae_geometricPointwiseBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/GeometricPointwiseBaseChangeTFAE.lean, line 50.
Stacks 0DKG. For a cartesian square of algebraic spaces,
K ∈ D_QCoh(𝒪_X),K' ∈ D_QCoh(𝒪_{X'})andφ : L(g')^*K ⟶ K', the following are equivalent: (1) the map of Equation 0DKF is an isomorphism at every geometric point ofX'in every degree; (2) for every diagram of schemesU ⟶ V ⟵ V'étale overX ⟶ Y ⟵ Y'withU' = V' ×_V U, the conditions of Stacks 0DJ8 hold forLa^*KandL(a')^*K'; (3) they hold for some such diagram withU' ⟶ X'surjective.
theorem tfae_geometricPointwiseBaseChange {X Y X' Y' : AlgebraicSpaces B S hS} {f : X ⟶ Y}
{g : Y' ⟶ Y} {f' : X' ⟶ Y'} {g' : X' ⟶ X} (sq : IsPullback g' f' f g)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hK : QcohDerived B S hS X K)
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(hK' : QcohDerived B S hS X' K')
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K') :
List.TFAE [GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ,
EtaleDiagramChartwiseBaseChange f g f' g' K hK K' hK' φ,
SurjectiveDiagramChartwiseBaseChange f g f' g' K hK K' hK' φ]
Tag 0DKH, a lemma
Tag 0DKH in the Stacks Project.
AlgebraicGeometry.Space.isIso_smallEtaleDerivedBaseChangeComp_of_geometricPointwiseBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/GeometricPointwiseBaseChangeIso.lean, line 44.
Stacks 0DKH. For a cartesian square of algebraic spaces with
fquasi-compact and quasi-separated,K ∈ D_QCoh(𝒪_X),K' ∈ D_QCoh(𝒪_{X'})andφ : L(g')^*K ⟶ K'satisfying the equivalent conditions of Stacks 0DKG (in the form of condition (1)), the compositeLg^*Rf_*K ⟶ Rf'_*L(g')^*K ⟶ Rf'_*K'is an isomorphism.
theorem isIso_smallEtaleDerivedBaseChangeComp_of_geometricPointwiseBaseChange
{X Y X' Y' : AlgebraicSpaces B S hS} {f : X ⟶ Y} {g : Y' ⟶ Y} {f' : X' ⟶ Y'}
{g' : X' ⟶ X} (sq : IsPullback g' f' f g) (hfqc : IsQuasiCompact B S hS f.hom)
(hfqs : IsQuasiSeparated B S hS f.hom)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hK : QcohDerived B S hS X K)
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(hK' : QcohDerived B S hS X' K')
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(h : GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ) :
IsIso (smallEtaleDerivedBaseChangeComp B S hS f g f' g' sq.w K K' φ)
Tag 0DKI, a lemma
Tag 0DKI in the Stacks Project.
AlgebraicGeometry.Space.geometricPointwiseBaseChange_tensor, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GeometricPointwiseBaseChangeTensorSheafHom.lean, line 84. The attribute's comment: “(1)”.
The conditions of Stacks 0DKG pass to derived tensor products (Stacks 0DKI (1)). For a cartesian square,
E,Kwith quasi-coherent cohomology onX,K'onX'andφ : L(g')^*K ⟶ K'satisfying the conditions of Stacks 0DKG, they hold forL(g')^*(E ⊗ᴸ K) ⟶ L(g')^*E ⊗ᴸ K', Stacks 07A4 followed byL(g')^*E ⊗ᴸ φ.
theorem geometricPointwiseBaseChange_tensor (sq : IsPullback g' f' f g)
(E K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : QcohDerived B S hS X E) (hK : QcohDerived B S hS X K)
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(hK' : QcohDerived B S hS X' K')
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(h : GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ) :
GeometricPointwiseBaseChange B S hS f g f' g' sq.w
(((smallEtaleDerivedTensor B S hS X).obj E).obj K)
(((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).obj K')
((smallEtaleDerivedPullbackTensorIso g').hom.app (E, K) ≫
((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).map φ)
AlgebraicGeometry.Space.geometricPointwiseBaseChange_sheafHom_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GeometricPointwiseBaseChangeTensorSheafHom.lean, line 111. The attribute's comment: “(2)”.
The conditions of Stacks 0DKG pass to the derived internal Hom out of a perfect object (Stacks 0DKI (2)). For a cartesian square,
Eperfect andKwith quasi-coherent cohomology onX,K'with quasi-coherent cohomology onX'andφ : L(g')^*K ⟶ K'satisfying the conditions of Stacks 0DKG, they hold forL(g')^*R𝓗om(E, K) ⟶ R𝓗om(L(g')^*E, K'), the map of Stacks 08JF followed byR𝓗om(L(g')^*E, φ).
theorem geometricPointwiseBaseChange_sheafHom_of_isPerfect (sq : IsPullback g' f' f g)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hK : QcohDerived B S hS X K)
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(hK' : QcohDerived B S hS X' K')
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(h : GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ) :
GeometricPointwiseBaseChange B S hS f g f' g' sq.w
(((smallEtaleDerivedSheafHom B S hS X).obj (op E)).obj K)
(((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).obj K')
(smallEtaleDerivedPullbackSheafHomMap g' E K ≫
((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).map φ)
AlgebraicGeometry.Space.geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/GeometricPointwiseBaseChangeTensorSheafHom.lean, line 144. The attribute's comment: “(3), corrected: g of finite Tor dimension added; (3) as written is false, see Stacks.Examples.PseudoCoherentSheafHomBaseChange”.
The conditions of Stacks 0DKG pass to the derived internal Hom out of a pseudo-coherent object, for
gof finite Tor dimension (Stacks 0DKI (3), corrected). Part (3) of the source has no condition ongand is false as written: seenot_forall_geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherentin the namespaceStacks.Examples.PseudoCoherentSheafHomBaseChange(A = ℤ/4,g : Spec ℤ/2 ⟶ Spec A, where the map of Equation 0DKF fails in degree1). The source proof commutes the infinite products ofHom^•(E_x, K_x)with⊗ᴸ R', which needsR'of finite Tor dimension. No boundedness ofK'is assumed: on every affine étale chart it follows from the conditions for(K, K', φ).
theorem geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherent (sq : IsPullback g' f' f g)
(hg : HasFiniteTorDimension g)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : RingedSite.IsPseudoCoherent (smallEtaleRingedSite B S hS X.1 X.2) E)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hK : DerivedCategoryB.derivedCategoryBPlus
(A := SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(SheafOfModules.isQuasicoherentOnSite.{u + 1, u, u, u}
((Stacks.SheafOfModules.Tensor.ringSheaf (smallEtaleRingedSite B S hS X.1 X.2).O))) K)
(K' : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X')))
(hK' : QcohDerived B S hS X' K')
(φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K')
(h : GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ) :
GeometricPointwiseBaseChange B S hS f g f' g' sq.w
(((smallEtaleDerivedSheafHom B S hS X).obj (op E)).obj K)
(((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).obj K')
(smallEtaleDerivedPullbackSheafHomMap g' E K ≫
((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).map φ)
Stacks.Examples.PseudoCoherentSheafHomBaseChange.not_forall_geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherent, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/Examples/Spaces/PseudoCoherentSheafHomBaseChange.lean, line 908. The attribute's comment: “(3) as written is false: the erratum's counterexample, A = Z/4, g : Spec Z/2 → Spec A”.
Stacks 0DKI (3) is false as written. It is not true that for every cartesian square
g' ≫ f = f' ≫ gof algebraic spaces,K ∈ D_QCoh(𝒪_X)bounded below andφ : L(g')^*K ⟶ K'inD_QCoh(𝒪_{X'})satisfying the conditions of Stacks 0DKG, and every pseudo-coherentE, the conditions of Stacks 0DKG hold forL(g')^*R𝓗om(E, K) ⟶ R𝓗om(L(g')^*E, K')(Stacks 08JF followed byR𝓗om(L(g')^*E, φ)). The counterexample hasY = S = Spec ℤ/4andg : Spec ℤ/2 ⟶ S, which has infinite Tor dimension:not_geometricPointwiseBaseChange_sheafHom_residueObj. The corrected statement, withgof finite Tor dimension, isAlgebraicGeometry.Space.geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherent.
theorem not_forall_geometricPointwiseBaseChange_sheafHom_of_isPseudoCoherent :
¬ ∀ (B : BigFppfSite.{0}) (S : Scheme.{0}) (hS : B.Contains S)
(X Y X' Y' : AlgebraicSpaces B S hS) (f : X ⟶ Y) (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g)
(K : DerivedCategory (SheafOfModules.{0} (ringedToposStructureRingSheaf B S hS X))),
QcohDerived B S hS X K →
∀ (K' : DerivedCategory (SheafOfModules.{0} (ringedToposStructureRingSheaf B S hS X'))),
QcohDerived B S hS X' K' →
∀ (φ : (smallEtaleDerivedModulePullback B S hS g').obj K ⟶ K'),
GeometricPointwiseBaseChange B S hS f g f' g' sq.w K K' φ →
DerivedCategoryB.IsBoundedBelow K →
∀ (E : DerivedCategory (SheafOfModules.{0} (ringedToposStructureRingSheaf B S hS X))),
RingedSite.IsPseudoCoherent (smallEtaleRingedSite B S hS X.1 X.2) E →
GeometricPointwiseBaseChange B S hS f g f' g' sq.w
(((smallEtaleDerivedSheafHom B S hS X).obj (op E)).obj K)
(((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).obj K')
(smallEtaleDerivedPullbackSheafHomMap g' E K ≫
((smallEtaleDerivedSheafHom B S hS X').obj
(op ((smallEtaleDerivedModulePullback B S hS g').obj E))).map φ)
Tag 0A1K, a lemma
Tag 0A1K in the Stacks Project.
AlgebraicGeometry.Space.isIso_smallEtaleDerivedBaseChangeComp_derivedTensor, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeIso.lean, line 117.
Stacks 0A1K. For a cartesian square of algebraic spaces with
fquasi-compact and quasi-separated,E ∈ D_QCoh(𝒪_X)and a bounded above complex𝒢^•of quasi-coherent𝒪_X-modules flat overY, the base change mapLg^*Rf_*(E ⊗^L 𝒢^•) ⟶ Rf'_*(L(g')^*E ⊗^L (g')^*𝒢^•)is an isomorphism.
theorem isIso_smallEtaleDerivedBaseChangeComp_derivedTensor
(sq : IsPullback g' f' f g) (hfqc : IsQuasiCompact B S hS f.hom)
(hfqs : IsQuasiSeparated B S hS f.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : QcohDerived B S hS X E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hGb : ∃ b : ℤ, ∀ i, b < i → IsZero (G.X i)) (hq : ∀ n, (G.X n).IsQuasicoherentOnSite)
(hfl : ∀ n, IsFlatOver B S hS X.2 f.hom (G.X n)) :
IsIso (smallEtaleDerivedBaseChangeComp B S hS f g f' g' sq.w
(((smallEtaleDerivedTensor B S hS X).obj E).obj (DerivedCategory.Q.obj G))
(((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).obj
(DerivedCategory.Q.obj (((modulePullback B S hS g').mapHomologicalComplex
(ComplexShape.up ℤ)).obj G)))
((smallEtaleDerivedPullbackTensorIso g').hom.app (E, DerivedCategory.Q.obj G) ≫
((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).map
(smallEtaleDerivedModulePullbackQComparison g' G)))
Tag 08JQ, a lemma
Tag 08JQ in the Stacks Project.
AlgebraicGeometry.Space.isIso_smallEtaleDerivedSheafHomBaseChangeMap_and_not_forall_isIso, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeSheafHomErratum.lean, line 79. The attribute's comment: “erratum: as printed, part (2) (E pseudo-coherent, G bounded and flat over Y, an arbitrary base change g) is false, refuted for X = Y = Spec Z/4, f = id, E = (Z/4/2)~, G = O_X and g the closed point Spec Z/2 -> Spec Z/4 (third conjunct); it holds when g has finite Tor dimension (second conjunct); part (1) holds as printed (first conjunct)”.
Let
f : X ⟶ Ybe a quasi-compact and quasi-separated morphism of algebraic spaces,E ∈ D(𝒪_X)and𝒢^•a complex of quasi-coherent𝒪_X-modules, each𝒢^nflat overY. For every cartesian square, the base change mapLg^*Rf_*R𝓗om(E, 𝒢^•) ⟶ Rf'_*R𝓗om(L(g')^*E, (g')^*𝒢^•)is an isomorphism if
Eis perfect and𝒢^•is bounded above (the source's part (1), at its hypotheses), orEis pseudo-coherent,𝒢^•is bounded, andghas finite Tor dimension.Part (2) as printed, for an arbitrary base change
g, is false, and the third conjunct proves it: it fails forX = Y = Spec ℤ/4,f = id,E = (ℤ/4 ⧸ 2)^~,𝒢^• = 𝒪_Xandg = g' : Spec ℤ/2 ⟶ Spec ℤ/4, where the source of the map lies inD^{≤ 0}(ℤ/4is self-injective) and the target has nonzero cohomology in degree1(not_forall_isIso_smallEtaleDerivedSheafHomBaseChangeMap_of_isPseudoCoherent, in the namespaceAlgebraicGeometry.Space.SheafHomBaseChangeCounterexample). The source proves (2) through Stacks 0DKH and 0DKI (3); Stacks 0DKI (3) needsgof finite Tor dimension, and with that condition the proof goes through (AlgebraicGeometry.Space.isIso_smallEtaleDerivedSheafHomBaseChangeMap_of_isPseudoCoherent). The morphismSpec ℤ/2 ⟶ Spec ℤ/4of the counterexample does not have finite Tor dimension.
theorem isIso_smallEtaleDerivedSheafHomBaseChangeMap_and_not_forall_isIso
(f : X ⟶ Y) (hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hq : ∀ n, (G.X n).IsQuasicoherentOnSite)
(hfl : ∀ n, IsFlatOver B S hS X.2 f.hom (G.X n)) :
(RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E →
(∃ b : ℤ, ∀ i, b < i → IsZero (G.X i)) →
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedSheafHomBaseChangeMap f g f' g' sq.w E G)) ∧
(RingedSite.IsPseudoCoherent (smallEtaleRingedSite B S hS X.1 X.2) E →
(∃ a b : ℤ, ∀ i, (i < a ∨ b < i) → IsZero (G.X i)) →
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g), HasFiniteTorDimension g →
IsIso (smallEtaleDerivedSheafHomBaseChangeMap f g f' g' sq.w E G)) ∧
¬ ∀ {B : BigFppfSite.{0}} {S : Scheme.{0}} {hS : B.Contains S} {X Y : AlgebraicSpaces B S hS}
(f : X ⟶ Y), IsQuasiCompact B S hS f.hom → IsQuasiSeparated B S hS f.hom →
∀ (E : DerivedCategory (SheafOfModules.{0} (ringedToposStructureRingSheaf B S hS X))),
RingedSite.IsPseudoCoherent (smallEtaleRingedSite B S hS X.1 X.2) E →
∀ (G : CochainComplex (SheafOfModules.{0} (ringedToposStructureRingSheaf B S hS X)) ℤ),
(∀ n, (G.X n).IsQuasicoherentOnSite) → (∀ n, IsFlatOver B S hS X.2 f.hom (G.X n)) →
(∃ a b : ℤ, ∀ i, (i < a ∨ b < i) → IsZero (G.X i)) →
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedSheafHomBaseChangeMap f g f' g' sq.w E G)
Sources
Section75.22 Producing perfect complexes#
Section 75.22 of the Stacks Project, Tag 0A1L. 3 tags here are carried by 3 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 08IS, a lemma
Tag 08IS in the Stacks Project.
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_of_hasProperSupport, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectDirectImage.lean, line 208. References: SGA 6 IV (2.12), “partial”.
Perfect direct images. Let
Sbe a scheme,Ya Noetherian algebraic space overSandf : X ⟶ Ya morphism of algebraic spaces which is locally of finite type and quasi-separated. LetE ∈ D(𝒪_X)be such that
E ∈ D^b_{Coh}(𝒪_X),- the support of
Hⁱ(E)is proper overYfor alli,Ehas finite tor dimension as an object ofD(f⁻¹𝒪_Y).Then
Rf_*Eis a perfect object ofD(𝒪_Y).
Rf_*is the total right derived functor of the direct image of modules, and the tor dimension ofE, read inD(f⁻¹𝒪_Y)by restriction of scalars alongf♯ : f⁻¹𝒪_Y ⟶ 𝒪_X, is measured against any derived tensor productTonf⁻¹𝒪_Y-modules (they all agree, being characterized by their values on K-flat complexes).
theorem isPerfect_derivedModulePushforward_of_hasProperSupport
(hTop : B.Contains (⊤_ Scheme.{u})) (f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom)
(hfqs : IsQuasiSeparated B S hS f.hom) (hY : IsNoetherian B S hS hTop Y.1)
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : derivedCategoryBBounded (isCoherent.{u, u} B S hS X.1 X.2) E)
(hsupp : ∀ i : ℤ, HasProperSupport B S hS X.2 Y.2 f.hom hf
((SheafOfModules.toSheaf _).obj
((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) i).obj E)))
(T : Stacks.SheafOfModules.DerivedTensorProduct.{u + 1, u, u, u + 1}
(inverseImageRingedSite B S hS f))
(hTor : Stacks.SheafOfModules.DerivedTensorProduct.HasFiniteTorDimension.{u + 1, u, u, u + 1} T
((derivedScalarRestriction B S hS f).obj E)) :
(smallEtaleRingedSite B S hS Y.1 Y.2).IsPerfect
((Stacks.DerivedFunctor.totalRightDerived (modulePushforward B S hS f)).obj E)
Tag 0DKJ, a lemma
Tag 0DKJ in the Stacks Project.
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_derivedTensor, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectPushforwardTensor.lean, line 90.
Stacks 0DKJ. Let
Ybe a Noetherian algebraic space andf : X ⟶ Ya morphism locally of finite type and quasi-separated. LetEbe a perfect object ofD(𝒪_X)and𝒢^•a bounded complex of coherent𝒪_X-modules, flat overY, with support proper overY. ThenRf_*(E ⊗^L_{𝒪_X} 𝒢^•)is a perfect object ofD(𝒪_Y), for any derived tensor product⊗^LonD(𝒪_X).
theorem isPerfect_derivedModulePushforward_derivedTensor (hTop : B.Contains (⊤_ Scheme.{u}))
(f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1)
(T : Stacks.SheafOfModules.DerivedTensorProduct.{u + 1, u, u, u + 1}
(smallEtaleRingedSite B S hS X.1 X.2))
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E)
(G : CochainComplex (Modules.{u, u} B S hS X.1 X.2) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) :
(smallEtaleRingedSite B S hS Y.1 Y.2).IsPerfect
((Stacks.DerivedFunctor.totalRightDerived (modulePushforward B S hS f)).obj
((T.tensor.obj E).obj (DerivedCategory.Q.obj G)))
Tag 0DKK, a lemma
Tag 0DKK in the Stacks Project.
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_derivedSheafHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/PerfectPushforwardInternalHom.lean, line 62.
Stacks 0DKK. Let
Ybe a Noetherian algebraic space andf : X ⟶ Ya morphism locally of finite type and quasi-separated. LetEbe a perfect object ofD(𝒪_X)and𝒢^•a bounded complex of coherent𝒪_X-modules, flat overY, with support proper overY. ThenRf_* RSheafHom(E, 𝒢^•)is a perfect object ofD(𝒪_Y).
theorem isPerfect_derivedModulePushforward_derivedSheafHom (hTop : B.Contains (⊤_ Scheme.{u}))
(f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1)
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E)
(G : CochainComplex (Modules.{u, u} B S hS X.1 X.2) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) :
(smallEtaleRingedSite B S hS Y.1 Y.2).IsPerfect
((Stacks.DerivedFunctor.totalRightDerived (modulePushforward B S hS f)).obj
(((RingedSite.derivedSheafHomSmallBifunctor (smallEtaleRingedSite B S hS X.1 X.2)).obj
(op E)).obj (DerivedCategory.Q.obj G)))
Sources
Section75.23 A projection formula for Ext#
Section 75.23 of the Stacks Project, Tag 08JM. 4 tags here are carried by 5 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0A1M, a lemma
Tag 0A1M in the Stacks Project.
AlgebraicGeometry.Space.exists_cohomologyTensorDirectImage_iso, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyTensorPullbackComplex.lean, line 721.
Stacks 0A1M. Let
Ybe a Noetherian algebraic space andf : X ⟶ Ya morphism locally of finite type and quasi-separated. LetEbe a perfect object ofD(𝒪_X)and𝒢^•a bounded complex of coherent𝒪_X-modules, flat overY, with support proper overY, andK = Rf_*(E ⊗^L 𝒢^•)(Stacks 0DKJ). Then there are mapsH^i(Y, K ⊗^L ℱ) ⟶ H^i(X, E ⊗^L (𝒢^• ⊗ f^*ℱ)), natural in the𝒪_Y-moduleℱ, which are isomorphisms forℱquasi-coherent, and which are compatible with the boundary maps: for a short exact sequence0 → ℱ₁ → ℱ₂ → ℱ₃ → 0the boundary maps of the triangleK ⊗^L ℱ₁ → K ⊗^L ℱ₂ → K ⊗^L ℱ₃ → K ⊗^L ℱ₁[1]correspond to those of the triangle of the short exact sequence of complexes0 → 𝒢^• ⊗ f^*ℱ₁ → 𝒢^• ⊗ f^*ℱ₂ → 𝒢^• ⊗ f^*ℱ₃ → 0tensored withE. Here⊗^Lis the canonical derived tensor product on the small étale site.
theorem exists_cohomologyTensorDirectImage_iso :
∃ η : ∀ i : ℤ, cohomologyTensorDirectImage B S hS f E G i ⟶
cohomologyTensorPullbackComplex B S hS f E G i,
(∀ (i : ℤ) (F : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)),
F.IsQuasicoherentOnSite → IsIso ((η i).app F)) ∧
∀ {T : ShortComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))}
(hT : T.ShortExact) (i j : ℤ) (h : i + 1 = j),
(forget₂ (ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS Y.1 Y.2))
AddCommGrpCat.{u + 1}).map
((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS Y.1 Y.2)) 0).homologySequenceδ
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS Y.1 Y.2).mapTriangle.obj
(((smallEtaleDerivedTensor B S hS Y).obj
(tensorDirectImageObj B S hS f E G)).mapTriangle.obj
(DerivedCategory.triangleOfSES
(hT.map_of_exact (HomologicalComplex.single (SheafOfModules.{u}
(ringedToposStructureRingSheaf B S hS Y)) (ComplexShape.up ℤ) 0)))))
i j h) ≫ (η j).app T.X₁ =
(η i).app T.X₃ ≫
(forget₂ (ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2))
AddCommGrpCat.{u + 1}).map
((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2)) 0).homologySequenceδ
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).mapTriangle.obj
(((smallEtaleDerivedTensor B S hS X).obj E).mapTriangle.obj
(DerivedCategory.triangleOfSES (shortExact_tensorPullbackComplex_map_single
((FlatProperSupportComplex.isFlatOverAtGeometricPoint B S hS hG))
hT))))
i j h)
Tag 08JN, a lemma
Tag 08JN in the Stacks Project.
AlgebraicGeometry.Space.exists_extPerfectCohomologyIso, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ExtPerfectDirectImageCoefficient.lean, line 91.
Stacks 08JN. Let
Ybe a Noetherian algebraic space andf : X ⟶ Ya morphism locally of finite type and quasi-separated, letE ∈ D(𝒪_X)be perfect and𝒢^•a bounded complex of coherent𝒪_X-modules, flat overY, with support proper overY(the notation of Stacks 0DKK), and letK = Rf_* RSheafHom(E, 𝒢^•). Then there are isomorphismsH^i(Y, K ⊗^L ℱ) ≅ Ext^i_{𝒪_X}(E, 𝒢^• ⊗ f^*ℱ)for quasi-coherent𝒪_Y-modulesℱ, functorial inℱ, and compatible with the boundary maps: for a short exact sequence0 → ℱ₁ → ℱ₂ → ℱ₃ → 0of quasi-coherent𝒪_Y-modules the boundary mapH^i(Y, K ⊗^L ℱ₃) → H^{i+1}(Y, K ⊗^L ℱ₁)corresponds to the boundary mapExt^i(E, 𝒢^• ⊗ f^*ℱ₃) → Ext^{i+1}(E, 𝒢^• ⊗ f^*ℱ₁)of the short exact sequence of complexes0 → 𝒢^• ⊗ f^*ℱ₁ → 𝒢^• ⊗ f^*ℱ₂ → 𝒢^• ⊗ f^*ℱ₃ → 0.Here
K = Rf_* RSheafHom(E, 𝒢^•)is formed with the internal homAlgebraicGeometry.Space.smallEtaleDerivedSheafHomofD(𝒪_X)(the one adjoint to the derived tensor product of the small étale site, Stacks 08J8); it is perfect byAlgebraicGeometry.Space.isPerfect_derivedModulePushforward_smallEtaleDerivedSheafHom. Stacks 0DKK itself is registered, on the internal hom of the ringed site, asAlgebraicGeometry.Space.isPerfect_derivedModulePushforward_derivedSheafHom.H^i(Y, -)is thei-th cohomology of derived global sections, andExt^i(E, -) = Hom(E, -[i])inD(𝒪_X).
theorem exists_extPerfectCohomologyIso (hTop : B.Contains (⊤_ Scheme.{u})) (f : X ⟶ Y)
(hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) :
∃ c : ∀ (F : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)),
F.IsQuasicoherentOnSite → ∀ i : ℤ,
extPerfectBaseCohomology B S hS f E G F i ≅ extPerfectExtGroup B S hS f E G F i,
(∀ (F F' : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(hF : F.IsQuasicoherentOnSite) (hF' : F'.IsQuasicoherentOnSite) (q : F ⟶ F') (i : ℤ),
(forget₂ (ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS Y.1 Y.2))
AddCommGrpCat.{u + 1}).map (((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS Y.1 Y.2)) 0).shift i).map
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS Y.1 Y.2).map
(((smallEtaleDerivedTensor B S hS Y).obj
(extPerfectDirectImageObj B S hS f E G)).map (extPerfectSingleMap B S hS q)))) ≫
(c F' hF' i).hom =
(c F hF i).hom ≫
(preadditiveCoyoneda.obj (op E)).map ((extPerfectCoefficientMap B S hS f G q)⟦i⟧')) ∧
(∀ (T : ShortComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)))
(hT : T.ShortExact) (h₁ : T.X₁.IsQuasicoherentOnSite) (h₃ : T.X₃.IsQuasicoherentOnSite)
(i j : ℤ) (h : i + 1 = j),
extPerfectBaseDelta B S hS f E G T hT i j h ≫ (c T.X₁ h₁ j).hom =
(c T.X₃ h₃ i).hom ≫ (preadditiveCoyoneda.obj (op E)).map
((DerivedCategory.triangleOfSESδ (shortExact_tensorPullbackComplex_map_single
(hG.isFlatOverAtGeometricPoint B S hS) hT))⟦i⟧' ≫
(shiftFunctorAdd' _ 1 i j (by omega)).inv.app
(extPerfectCoefficientObj B S hS f G T.X₁)))
Tag 08JR, a lemma
Tag 08JR in the Stacks Project.
AlgebraicGeometry.Space.exists_isPseudoCoherent_extAddEquiv_tensorPullbackComplex, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ExtPerfectDualTowerData.lean, line 705. The attribute's comment: “(B)”.
Stacks 08JR, part (B). Let
Sbe a scheme andf : X ⟶ Ba morphism of algebraic spaces overSwithBNoetherian andflocally of finite type and quasi-separated. LetE ∈ D^-_{Coh}(𝒪_X)(coherent cohomology sheaves, bounded above) and let𝒢^•be a bounded complex of coherent𝒪_X-modules, flat overB, with support proper overB. Then there is a pseudo-coherentL ∈ D(𝒪_B)with isomorphismsExt^i_{𝒪_B}(L, ℱ) ≅ Ext^i_{𝒪_X}(E, 𝒢^• ⊗_{𝒪_X} f^*ℱ)for quasi-coherent𝒪_B-modulesℱ, functorial inℱand compatible with the boundary maps: for a short exact sequence0 → ℱ₁ → ℱ₂ → ℱ₃ → 0of quasi-coherent modules, the boundary map of the distinguished triangle ofℱ₁[0] → ℱ₂[0] → ℱ₃[0]on the left corresponds to the boundary map of the distinguished triangle of the short exact sequence of complexes0 → 𝒢^• ⊗ f^*ℱ₁ → 𝒢^• ⊗ f^*ℱ₂ → 𝒢^• ⊗ f^*ℱ₃ → 0on the right.Here
Ext^i(A, C) = Hom(A, C[i])in the derived categories of the small étale ringed sites,ℱ[0]isDerivedCategory.singleFunctor _ 0, and𝒢^• ⊗ f^*ℱisAlgebraicGeometry.Space.tensorPullbackComplexatℱplaced in degree0; the sequence of complexes is writtenShortComplex.mk (ℱ₁[0] → ℱ₂[0]) (ℱ₂[0] → ℱ₃[0])mapped by𝒢^• ⊗ f^*(-), exact byAlgebraicGeometry.Space.shortExact_tensorPullbackComplex_map_single.
theorem exists_isPseudoCoherent_extAddEquiv_tensorPullbackComplex (hTop : B.Contains
(⊤_ Scheme.{u}))
(f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1) (E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hEcoh : ∀ q : ℤ, IsCoherent B S hS X.1 X.2 ((DerivedCategory.homologyFunctor _ q).obj E))
(hEb : DerivedCategoryB.IsBoundedAbove E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) :
∃ (L : DerivedCategory (Modules.{u, u} B S hS Y.1 Y.2))
(_ : (smallEtaleRingedSite B S hS Y.1 Y.2).IsPseudoCoherent L)
(Φ : ∀ (F : Modules.{u, u} B S hS Y.1 Y.2) (_ : F.IsQuasicoherentOnSite) (i : ℤ),
(L ⟶ ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).obj F)⟦i⟧) ≃+
ShiftedHom E (tensorPullbackComplexDerived f G ((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(ComplexShape.up ℤ) 0).obj F)) i),
(∀ {F F' : Modules.{u, u} B S hS Y.1 Y.2} (hF : F.IsQuasicoherentOnSite)
(hF' : F'.IsQuasicoherentOnSite) (q : F ⟶ F') (i : ℤ)
(e : L ⟶ ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).obj F)⟦i⟧),
Φ F' hF' i (e ≫ ((DerivedCategory.singleFunctor
(Modules.{u, u} B S hS Y.1 Y.2) 0).map q)⟦i⟧') =
Φ F hF i e ≫ (DerivedCategory.Q.map ((tensorPullbackComplex f G).map
((HomologicalComplex.single (SheafOfModules.{u}
(ringedToposStructureRingSheaf B S hS Y)) (ComplexShape.up ℤ) 0).map q)))⟦i⟧') ∧
(∀ (T : ShortComplex (Modules.{u, u} B S hS Y.1 Y.2)) (hT : T.ShortExact)
(h₁ : T.X₁.IsQuasicoherentOnSite) (h₂ : T.X₂.IsQuasicoherentOnSite)
(h₃ : T.X₃.IsQuasicoherentOnSite) (i : ℤ)
(e : ShiftedHom L ((DerivedCategory.singleFunctor
(Modules.{u, u} B S hS Y.1 Y.2) 0).obj T.X₃) i),
Φ T.X₁ h₁ (i + 1) (e.comp hT.singleδ (add_comm 1 i)) =
(Φ T.X₃ h₃ i e).comp (DerivedCategory.triangleOfSESδ
(S := (ShortComplex.mk ((HomologicalComplex.single (SheafOfModules.{u}
(ringedToposStructureRingSheaf B S hS Y)) (ComplexShape.up ℤ) 0).map T.f)
((HomologicalComplex.single (SheafOfModules.{u}
(ringedToposStructureRingSheaf B S hS Y)) (ComplexShape.up ℤ) 0).map T.g)
(single_map_comp_map_eq_zero B S hS T)).map (tensorPullbackComplex f G))
(shortExact_tensorPullbackComplex_map_single
(FlatProperSupportComplex.isFlatOverAtGeometricPoint B S hS hG) hT))
(add_comm 1 i))
AlgebraicGeometry.Space.exists_isPerfect_tensorPullbackExtGroup_smallEtaleTensorCohomology, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ExtTensorPullbackPerfectApproximation.lean, line 527. The attribute's comment: “(A)”.
Stacks 08JR, part (A). Let
Sbe a scheme andf : X ⟶ Ba morphism of algebraic spaces overSwithBNoetherian andflocally of finite type and quasi-separated. LetE ∈ D^-_{Coh}(𝒪_X)(coherent cohomology sheaves, bounded above) and let𝒢^•be a bounded complex of coherent𝒪_X-modules, flat overB, with support proper overB. Then for everym ∈ ℤthere are a perfectK ∈ D(𝒪_B)and mapsα^i_ℱ : Ext^i_{𝒪_X}(E, 𝒢^• ⊗_{𝒪_X} f^*ℱ) ⟶ H^i(B, K ⊗^L_{𝒪_B} ℱ)for quasi-coherent𝒪_B-modulesℱ, functorial inℱ, compatible with the boundary maps of short exact sequences0 → ℱ₁ → ℱ₂ → ℱ₃ → 0of quasi-coherent modules (on the left the boundary map of the distinguished triangle of0 → 𝒢^• ⊗ f^*ℱ₁ → 𝒢^• ⊗ f^*ℱ₂ → 𝒢^• ⊗ f^*ℱ₃ → 0, on the right the connecting map of the cohomology ofRΓ(B, K ⊗^L -)of the triangle ofℱ₁[0] → ℱ₂[0] → ℱ₃[0]), and invertible fori ≤ m.Here
Ext^i(E, -) = Hom(E, -[i])inD(𝒪_X)(AlgebraicGeometry.Space.tensorPullbackExtGroup),H^i(B, K ⊗^L ℱ)is thei-th cohomology of derived global sections ofK ⊗^L ℱ[0](AlgebraicGeometry.Space.smallEtaleTensorCohomology), and the maps are morphisms of abelian groups.
theorem exists_isPerfect_tensorPullbackExtGroup_smallEtaleTensorCohomology
(hTop : B.Contains (⊤_ Scheme.{u}))
(f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1) (E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hEcoh : ∀ q : ℤ, IsCoherent B S hS X.1 X.2 ((DerivedCategory.homologyFunctor _ q).obj E))
(hEb : DerivedCategoryB.IsBoundedAbove E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) (m : ℤ) :
∃ (K : smallEtaleModulesDerived B S hS Y) (_ : SmallEtaleIsPerfect B S hS K)
(α : ∀ (F : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(_ : F.IsQuasicoherentOnSite) (i : ℤ),
tensorPullbackExtGroup B S hS E f G F i ⟶ smallEtaleTensorCohomology B S hS K F i),
(∀ (F : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(hF : F.IsQuasicoherentOnSite) (i : ℤ), i ≤ m → IsIso (α F hF i)) ∧
(∀ {F F' : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)}
(hF : F.IsQuasicoherentOnSite) (hF' : F'.IsQuasicoherentOnSite) (q : F ⟶ F') (i : ℤ),
tensorPullbackExtGroupMap B S hS E f G q i ≫ α F' hF' i =
α F hF i ≫ smallEtaleTensorCohomologyMap B S hS K q i) ∧
(∀ (T : ShortComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)))
(hT : T.ShortExact) (h₁ : T.X₁.IsQuasicoherentOnSite) (h₃ : T.X₃.IsQuasicoherentOnSite)
(i j : ℤ) (h : i + 1 = j),
tensorPullbackExtGroupDelta B S hS E f G T (shortExact_tensorPullbackComplex_map_single
(FlatProperSupportComplex.isFlatOverAtGeometricPoint B S hS hG) hT) i j h ≫
α T.X₁ h₁ j =
α T.X₃ h₃ i ≫ smallEtaleTensorCohomologyDelta B S hS K T hT i j h)
Tag 0DKL, a remark
Tag 0DKL in the Stacks Project.
AlgebraicGeometry.Space.exists_isPseudoCoherent_extAddEquiv_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ExtRepresentingComplexBaseChange.lean, line 126.
Stacks 0DKL. Let
Ybe a Noetherian algebraic space,f : X ⟶ Ylocally of finite type and quasi-separated,E ∈ D^-_{Coh}(𝒪_X)and𝒢^•a bounded complex of coherent𝒪_X-modules, flat overY, with support proper overY. The pseudo-coherentL ∈ D(𝒪_Y)of Stacks 08JR (B) has isomorphismsExt^i_Y(L, ℱ) ≅ Ext^i_X(E, 𝒢^• ⊗ f^*ℱ), natural in the quasi-coherentℱand compatible with boundary maps. It is compatible with base change: for every cartesian squareX' = X ×_Y Y'(withg : Y' ⟶ Yarbitrary) there are isomorphismsExt^i_{Y'}(Lg^*L, ℱ') ≅ Ext^i_{X'}(L(g')^*E, (g')^*𝒢^• ⊗ (f')^*ℱ')forℱ'quasi-coherent onY', natural inℱ'. Here(g')^*𝒢^•is the termwise, underived, pullback.
theorem exists_isPseudoCoherent_extAddEquiv_baseChange (hTop : B.Contains (⊤_ Scheme.{u}))
(f : X ⟶ Y) (hf : IsLocallyOfFiniteType B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hY : IsNoetherian B S hS hTop Y.1) (E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hEcoh : ∀ q : ℤ, IsCoherent B S hS X.1 X.2 ((DerivedCategory.homologyFunctor _ q).obj E))
(hEb : DerivedCategoryB.IsBoundedAbove E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hG : FlatProperSupportComplex B S hS f hf G) :
∃ (L : DerivedCategory (Modules.{u, u} B S hS Y.1 Y.2))
(_ : (smallEtaleRingedSite B S hS Y.1 Y.2).IsPseudoCoherent L)
(Φ : ∀ (F : Modules.{u, u} B S hS Y.1 Y.2) (_ : F.IsQuasicoherentOnSite) (i : ℤ),
(L ⟶ ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).obj F)⟦i⟧) ≃+
ShiftedHom E (tensorPullbackComplexDerived f G ((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)) (ComplexShape.up ℤ) 0).obj
F)) i),
(∀ {F F' : Modules.{u, u} B S hS Y.1 Y.2} (hF : F.IsQuasicoherentOnSite)
(hF' : F'.IsQuasicoherentOnSite) (q : F ⟶ F') (i : ℤ)
(e : L ⟶ ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).obj F)⟦i⟧),
Φ F' hF' i (e ≫ ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).map
q)⟦i⟧') =
Φ F hF i e ≫ (DerivedCategory.Q.map ((tensorPullbackComplex f G).map
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(ComplexShape.up ℤ) 0).map q)))⟦i⟧') ∧
(∀ (T : ShortComplex (Modules.{u, u} B S hS Y.1 Y.2)) (hT : T.ShortExact)
(h₁ : T.X₁.IsQuasicoherentOnSite) (_h₂ : T.X₂.IsQuasicoherentOnSite)
(h₃ : T.X₃.IsQuasicoherentOnSite) (i : ℤ)
(e : ShiftedHom L ((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y.1 Y.2) 0).obj
T.X₃) i),
Φ T.X₁ h₁ (i + 1) (e.comp hT.singleδ (add_comm 1 i)) =
(Φ T.X₃ h₃ i e).comp (DerivedCategory.triangleOfSESδ (S := (ShortComplex.mk
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(ComplexShape.up ℤ) 0).map T.f)
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y))
(ComplexShape.up ℤ) 0).map T.g) (single_map_comp_map_eq_zero B S hS T)).map
(tensorPullbackComplex f G))
(shortExact_tensorPullbackComplex_map_single
(FlatProperSupportComplex.isFlatOverAtGeometricPoint B S hS hG) hT))
(add_comm 1 i)) ∧
∀ {X' Y' : AlgebraicSpaces B S hS} {g : Y' ⟶ Y} {g' : X' ⟶ X} {f' : X' ⟶ Y'}
(_ : IsPullback g' f' f g),
∃ Ψ : ∀ (F' : Modules.{u, u} B S hS Y'.1 Y'.2) (_ : F'.IsQuasicoherentOnSite) (i : ℤ),
(asModulesObj Y' ((smallEtaleDerivedModulePullback B S hS g).obj L) ⟶
((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y'.1 Y'.2) 0).obj F')⟦i⟧) ≃+
(asModulesObj X' ((smallEtaleDerivedModulePullback B S hS g').obj E) ⟶
(tensorPullbackComplexDerived f' ((pullbackComplexSmallEtale g').obj G)
((CochainComplex.singleFunctor _ 0).obj F'))⟦i⟧),
∀ {F' F'' : Modules.{u, u} B S hS Y'.1 Y'.2} (hF' : F'.IsQuasicoherentOnSite)
(hF'' : F''.IsQuasicoherentOnSite) (q : F' ⟶ F'') (i : ℤ)
(e : asModulesObj Y' ((smallEtaleDerivedModulePullback B S hS g).obj L) ⟶
((DerivedCategory.singleFunctor (Modules.{u, u} B S hS Y'.1 Y'.2) 0).obj F')⟦i⟧),
Ψ F'' hF'' i (e ≫ ((DerivedCategory.singleFunctor _ 0).map q)⟦i⟧') =
Ψ F' hF' i e ≫ (DerivedCategory.Q.map ((tensorPullbackComplex f'
((pullbackComplexSmallEtale g').obj G)).map
((CochainComplex.singleFunctor _ 0).map q)))⟦i⟧'
Sources
Section75.24 Limits and derived categories#
Section 75.24 of the Stacks Project, Tag 09RG. 3 tags here are carried by 10 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 09RH, a situation
Tag 09RH in the Stacks Project.
AlgebraicGeometry.Space.DescentSituation, a structure in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/Limits.lean, line 53.
Stacks 09RH. A directed inverse system of quasi-compact and quasi-separated algebraic spaces over
Swhose transition morphisms are affine, equipped with a chosen limit cone and a chosen index.
structure DescentSituation (B : BigFppfSite.{u}) (S : Scheme.{u}) (hS : B.Contains S)
(hZ : B.Contains (⊤_ Scheme.{u})) (I : Type v) [Preorder I] [IsDirected I (· ≤ ·)]
Tag 09RI, a lemma
Tag 09RI in the Stacks Project.
AlgebraicGeometry.Space.DescentSituation.isHomColimit_of_ringModel, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitDerivedHom.lean, line 420. The attribute's comment: “the affine ring-translation dictionary model is carried as an explicit hypothesis: the reduction of the sheaf-level statement to the ring-level colimit computation of Stacks 09RE, which runs through the induction principle 08GP, Mayer-Vietoris 08H9 and the small-etale-to-Zariski comparison 071Q, is not available in this library”.
Stacks 09RI. In Situation 09RH, let
E₀andK₀be objects ofD(𝒪_{X₀}), putE_i = Lf_{i0}^*E₀,K_i = Lf_{i0}^*K₀fori ≥ 0andE = Lf_0^*E₀,K = Lf_0^*K₀. If eitherE₀is perfect andK₀lies inD_QCoh(𝒪_{X₀}), orE₀is pseudo-coherent andK₀lies inD_QCoh(𝒪_{X₀})and has finite Tor dimension, then
colim_{i ≥ 0} Hom_{D(𝒪_{X_i})}(E_i, K_i) ⟶ Hom_{D(𝒪_X)}(E, K)is an isomorphism.
The affine translation dictionary
modelis carried as an explicit hypothesis rather than discharged; see "The carried hypothesis" in this file's module documentation for what it is and why it is an input of the source's proof and not its conclusion.The three clauses are read off the ring-level colimit computation of Stacks 09RE, which this library proves: surjectivity is
Stacks.FilteredColimitHom.exists_descends_derived, eventual injectivity isStacks.FilteredColimitHom.exists_le_eq_of_eq_derived, and the cocone identity is the compatibility of the dictionary withRingModel.hcompat.
theorem isHomColimit_of_ringModel
[∀ U : (smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property).C,
HasSheafify ((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).J.over U) AddCommGrpCat.{u}]
[∀ U : (smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property).C,
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).J.over U).WEqualsLocallyBijective AddCommGrpCat.{u}]
[∀ U : (smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property).C,
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).J.over U).HasSheafCompose
(forget₂ RingCat.{u} AddCommGrpCat.{u})]
[(Stacks.SheafOfModules.Tensor.sheafTensor.{u + 1, u, u, u, u}
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property).O).Additive]
[∀ M, ((Stacks.SheafOfModules.Tensor.sheafTensor.{u + 1, u, u, u, u}
(smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).O).obj M).Additive]
[∀ (M N : CochainComplex
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property) ℤ),
HomologicalComplex.HasMapBifunctor M N
(Stacks.SheafOfModules.Tensor.sheafTensor.{u + 1, u, u, u, u}
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property).O)
(ComplexShape.up ℤ)]
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(hcase :
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).IsPerfect E₀ ∧
∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property)
n).obj K₀).IsQuasicoherentOnSite) ∨
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).IsPseudoCoherent E₀ ∧
(∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property)
n).obj K₀).IsQuasicoherentOnSite) ∧
Stacks.SheafOfModules.DerivedTensorProduct.HasFiniteTorDimension T K₀))
(R : Type u) [Ring R] [HasDerivedCategory.{w} (ModuleCat.{u} R)]
(model : RingModel sys E₀ K₀ R) :
IsHomColimit sys E₀ K₀
AlgebraicGeometry.Space.DescentSituation.isHomColimit_canonicalDerivedPullbackSystem_of_isPerfect_or_isPseudoCoherent_of_mk_lt_cof_ord, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitDerivedHomPerfectQcohLeaf.lean, line 110. The attribute's comment: “both clauses, at the tag's own hypotheses”.
Stacks 09RI. In Situation 09RH, let
E₀andK₀be objects ofD(𝒪_{X₀}), putE_i = Lf_{i0}^*E₀,K_i = Lf_{i0}^*K₀fori ≥ 0andE = Lf_0^*E₀,K = Lf_0^*K₀. If eitherE₀is perfect andK₀lies inD_QCoh(𝒪_{X₀}), orE₀is pseudo-coherent andK₀lies inD_QCoh(𝒪_{X₀})and has finite Tor dimension, then
colim_{i ≥ 0} Hom_{D(𝒪_{X_i})}(E_i, K_i) ⟶ Hom_{D(𝒪_X)}(E, K)is an isomorphism.
This is
isHomColimit_canonicalDerivedPullbackSystem_of_isPerfect_or_isPseudoCoherent_of_qcohUnitwith its hypothesishunitdischarged byshiftedIsColim_baseChangeSystem_unit_of_isQuasicoherentOnSite, so both clauses of the tag are proved at its own hypotheses.The two restrictions beyond the tag are counterexample-forced for this formalization:
I : Type uis refuted beyond it byStacksFrontier/AlgebraicGeometry/Perfect/LimitHomDescent/Counterexample.lean, and#I < B.cutoff.ord.cofbyStacksFrontier/AlgebraicGeometry/Spaces/Limits/SectionsColimitLargeIndexCounterexample.lean.Tis the derived tensor product the Tor condition of clause (2) is predicated on, not an added assumption.
theorem isHomColimit_canonicalDerivedPullbackSystem_of_isPerfect_or_isPseudoCoherent_of_mk_lt_cof_ord
(E₀ K₀ : derivedModules B S hS (D.stage D.base))
(hI : Cardinal.mk I < B.cutoff.ord.cof)
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(hcase :
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).IsPerfect E₀ ∧
∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property)
n).obj K₀).IsQuasicoherentOnSite) ∨
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).IsPseudoCoherent E₀ ∧
(∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property)
n).obj K₀).IsQuasicoherentOnSite) ∧
Stacks.SheafOfModules.DerivedTensorProduct.HasFiniteTorDimension T K₀)) :
IsHomColimit D.canonicalDerivedPullbackSystem E₀ K₀
AlgebraicGeometry.Space.DescentSituation.isHomColimit_canonicalDerivedPullbackSystem_of_isPseudoCoherent_of_hasFiniteTorDimension, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitDerivedHomTorBridge09RI.lean, line 140. The attribute's comment: “clause (2); clause (1) not included”.
The pseudo-coherent case of Stacks 09RI at the spelling of the tag, in the
IsHomColimitform of the tag's conclusion:Hom_{D(𝒪_X)}(E, K)is the filtered colimit of theHom_{D(𝒪_{X_i})}(E_i, K_i)for the system of derived pullbacks of Situation 09RH with the canonical comparisons.This carries
@[stacks 09RI]for clause (2) of the tag —E₀pseudo-coherent, every cohomology sheaf ofK₀quasi-coherent,K₀of finite Tor dimension — at exactly those hypotheses, which is one half of the tag's covering set. Clause (1),E₀perfect withK₀only inD_QCoh, is not included here and is not a consequence of Stacks 0DKV.The two restrictions beyond the tag are counterexample-forced for this formalization, not conveniences:
I : Type uis refuted beyond it byStacksFrontier/AlgebraicGeometry/Perfect/LimitHomDescent/Counterexample.lean, and#I < B.cutoff.ord.cofbyStacksFrontier/AlgebraicGeometry/Spaces/Limits/SectionsColimitLargeIndexCounterexample.lean.Tis the derived tensor product the Tor condition is predicated on, not an added assumption.
theorem isHomColimit_canonicalDerivedPullbackSystem_of_isPseudoCoherent_of_hasFiniteTorDimension
(hI : Cardinal.mk I < B.cutoff.ord.cof)
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(hcase :
((smallEtaleRingedSite B S hS (D.stage D.base).obj
(D.stage D.base).property).IsPseudoCoherent E₀ ∧
(∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property)
n).obj K₀).IsQuasicoherentOnSite) ∧
Stacks.SheafOfModules.DerivedTensorProduct.HasFiniteTorDimension T K₀)) :
IsHomColimit D.canonicalDerivedPullbackSystem E₀ K₀
Tag 09RJ, a lemma
Tag 09RJ in the Stacks Project.
AlgebraicGeometry.Space.DescentSituation.exists_descendedHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitPerfectObjects.lean, line 242. The attribute's comment: “fullness of the comparison out of the colimit of the categories of perfect objects, which the source's proof takes from Stacks 09RI; essential surjectivity, the other half of that proof, is not included”.
Fullness of the comparison out of the colimit, Stacks 09RJ. In Situation 09RH, let
E₀andK₀be perfect objects ofD(𝒪_{X₀})and putE_i = Lf_{i0}^*E₀,K_i = Lf_{i0}^*K₀fori ≥ 0andE = Lf_0^*E₀,K = Lf_0^*K₀. Then every morphismE ⟶ KofD(𝒪_X)is the image, under the map of Stacks 09RI, of a morphismE_i ⟶ K_iat some stagei ≥ 0.The categories of perfect objects are full subcategories of the ambient derived categories, so this is fullness of the comparison functor out of the colimit of the
D_perf(𝒪_{X_i}).This is the first half of the full faithfulness that the source's proof of Stacks 09RJ takes from Stacks 09RI, and it is deduced from Stacks 09RI here. The hypotheses
modelandhqcK₀are carried; see "The carried hypotheses" in this file's module documentation.
theorem exists_descendedHom
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(hqcK₀ : ∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property) n).obj
K₀.obj).IsQuasicoherentOnSite)
(R : Type u) [Ring R] [HasDerivedCategory.{w} (ModuleCat.{u} R)]
(model : RingModel sys E₀.obj K₀.obj R)
(φ : limitObj sys E₀.obj ⟶ limitObj sys K₀.obj) :
∃ (i : D.Stages) (g : stageObj sys E₀.obj i ⟶ stageObj sys K₀.obj i),
homProjection sys E₀.obj K₀.obj i g = φ
AlgebraicGeometry.Space.DescentSituation.eventually_map_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitPerfectObjects.lean, line 270. The attribute's comment: “faithfulness of the comparison out of the colimit of the categories of perfect objects, which the source's proof takes from Stacks 09RI; essential surjectivity, the other half of that proof, is not included”.
Faithfulness of the comparison out of the colimit, Stacks 09RJ. In Situation 09RH, with
E₀andK₀perfect objects ofD(𝒪_{X₀})as above, two morphismsE_i ⟶ K_iat a stagei ≥ 0whose imagesE ⟶ KinD(𝒪_X)agree already have equal images at some later stage.Together with
AlgebraicGeometry.Space.DescentSituation.exists_descendedHomthis says that the hom-sets ofD_perf(𝒪_X)between objects coming from the system are the filtered colimits of the hom-sets at the stages, which is full faithfulness of the comparison out of the colimit of theD_perf(𝒪_{X_i}).This is the second half of the full faithfulness that the source's proof of Stacks 09RJ takes from Stacks 09RI, and it is deduced from Stacks 09RI here. The hypotheses
modelandhqcK₀are carried; see "The carried hypotheses" in this file's module documentation.
theorem eventually_map_eq
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(hqcK₀ : ∀ n : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS (D.stage D.base).obj (D.stage D.base).property) n).obj
K₀.obj).IsQuasicoherentOnSite)
(R : Type u) [Ring R] [HasDerivedCategory.{w} (ModuleCat.{u} R)]
(model : RingModel sys E₀.obj K₀.obj R) (i : D.Stages)
(g g' : stageObj sys E₀.obj i ⟶ stageObj sys K₀.obj i)
(hgg : homProjection sys E₀.obj K₀.obj i g = homProjection sys E₀.obj K₀.obj i g') :
∃ (j : D.Stages) (h : i ≤ j), homTransition sys E₀.obj K₀.obj i j h g =
homTransition sys E₀.obj K₀.obj i j h g'
AlgebraicGeometry.Space.DescentSituation.exists_descendedHom_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitPerfectObjectsPerfectTarget.lean, line 116. The attribute's comment: “fullness of the comparison out of the colimit of the categories of perfect objects, which the source's proof takes from Stacks 09RI; essential surjectivity, the other half of that proof, is not included”.
Fullness of the comparison out of the colimit, Stacks 09RJ. In Situation 09RH, let
E₀andK₀be perfect objects ofD(𝒪_{X₀})and putE_i = Lf_{i0}^*E₀,K_i = Lf_{i0}^*K₀fori ≥ 0andE = Lf_0^*E₀,K = Lf_0^*K₀. Then every morphismE ⟶ KofD(𝒪_X)is the image, under the map of Stacks 09RI, of a morphismE_i ⟶ K_iat some stagei ≥ 0.This is fullness of the comparison functor out of the colimit of the
D_perf(𝒪_{X_i}).
theorem exists_descendedHom_of_isPerfect
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(R : Type u) [Ring R] [HasDerivedCategory.{w} (ModuleCat.{u} R)]
(model : RingModel sys E₀.obj K₀.obj R)
(φ : limitObj sys E₀.obj ⟶ limitObj sys K₀.obj) :
∃ (i : D.Stages) (g : stageObj sys E₀.obj i ⟶ stageObj sys K₀.obj i),
homProjection sys E₀.obj K₀.obj i g = φ
AlgebraicGeometry.Space.DescentSituation.eventually_map_eq_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/LimitPerfectObjectsPerfectTarget.lean, line 134. The attribute's comment: “faithfulness of the comparison out of the colimit of the categories of perfect objects, which the source's proof takes from Stacks 09RI; essential surjectivity, the other half of that proof, is not included”.
Faithfulness of the comparison out of the colimit, Stacks 09RJ. In Situation 09RH, with
E₀andK₀perfect objects ofD(𝒪_{X₀})as above, two morphismsE_i ⟶ K_iat a stagei ≥ 0whose imagesE ⟶ KinD(𝒪_X)agree already have equal images at some later stage.This is faithfulness of the comparison functor out of the colimit of the
D_perf(𝒪_{X_i}).
theorem eventually_map_eq_of_isPerfect
(T : Stacks.SheafOfModules.DerivedTensorProduct
(smallEtaleRingedSite B S hS (D.stage D.base).obj (D.stage D.base).property))
(R : Type u) [Ring R] [HasDerivedCategory.{w} (ModuleCat.{u} R)]
(model : RingModel sys E₀.obj K₀.obj R) (i : D.Stages)
(g g' : stageObj sys E₀.obj i ⟶ stageObj sys K₀.obj i)
(hgg : homProjection sys E₀.obj K₀.obj i g = homProjection sys E₀.obj K₀.obj i g') :
∃ (j : D.Stages) (h : i ≤ j), homTransition sys E₀.obj K₀.obj i j h g =
homTransition sys E₀.obj K₀.obj i j h g'
AlgebraicGeometry.Space.DescentSituation.perfectObjectsDescend_and_perfectHomsDescend, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/RestrictedPerfectDescentOverIntegers.lean, line 960. The attribute's comment: “Under the size bound ChartsBounded on the chart systems of the stages, which the source does not state (the sites are size-bounded models of the big fppf site).”.
Stacks 09RJ. In Situation 09RH (
X = lim Xᵢalong affine transition morphisms of quasi-compact quasi-separated algebraic spaces) the category of perfect objects ofD(𝒪_X)is the colimit of the categories of perfect objects ofD(𝒪_{Xᵢ}): every perfect object onXisLfᵢ^*Eᵢfor some stageiand some perfectEᵢ(AlgebraicGeometry.Space.DescentSituation.PerfectObjectsDescend), and for perfectEᵢ,KᵢonXᵢthe mapcolim_{j ≥ i} Hom(Lf_{ji}^*Eᵢ, Lf_{ji}^*Kᵢ) ⟶ Hom(Lfᵢ^*Eᵢ, Lfᵢ^*Kᵢ)is bijective (AlgebraicGeometry.Space.DescentSituation.PerfectHomsDescend).Correction. The statement carries the hypothesis
ChartsBounded(the size boundAlgebraicGeometry.Space.ChartsBoundedon the chart systems of the stages), which the source does not state: the sites here areu-small models of the big fppf site with a size cutoff, and the bound is what keeps the limits of the affine charts of the stages inside the site (AlgebraicGeometry.Space.ChartsBounded.sizeBound). It holds for instance when#Iis below the cofinality of the cutoff (AlgebraicGeometry.Space.chartsBounded_of_lt_cof_ord).
theorem perfectObjectsDescend_and_perfectHomsDescend (hb : ChartsBounded B S hS D.system) :
D.PerfectObjectsDescend ∧ D.PerfectHomsDescend
AlgebraicGeometry.Space.DescentSituation.perfectObjectsDescend_and_perfectHomsDescend_of_mk_lt_cof_ord, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/RestrictedPerfectDescentUnbounded.lean, line 143. The attribute's comment: “both halves, at the tag's own hypotheses”.
Stacks 09RJ at the hypotheses of Situation 09RH, with no bounded-charts hypothesis. In Situation 09RH, indexed by a directed set in the universe of the site with fewer elements than the cofinality of the site's size cutoff, the category of perfect objects of
D(𝒪_X)is the colimit of the categories of perfect objects ofD(𝒪_{Xᵢ}): essential surjectivity (AlgebraicGeometry.Space.DescentSituation.PerfectObjectsDescend) and full faithfulness (AlgebraicGeometry.Space.DescentSituation.PerfectHomsDescend).The size bound
AlgebraicGeometry.Space.ChartsBoundedthatAlgebraicGeometry.Space.DescentSituation.perfectObjectsDescend_and_perfectHomsDescendassumes, and that the source does not state, is discharged here byAlgebraicGeometry.Space.DescentSituation.chartsBounded_of_mk_lt_cof_ord. What is left is#I < B.cutoff.ord.cof, which is not a weakening: it is refuted beyond that bound byStacksFrontier/AlgebraicGeometry/Spaces/Limits/SectionsColimitLargeIndexCounterexample.lean, andI : Type uis refuted beyond it byStacksFrontier/AlgebraicGeometry/Perfect/LimitHomDescent/Counterexample.lean.This carries
@[stacks 09RJ]for both halves — essential surjectivity (AlgebraicGeometry.Space.DescentSituation.PerfectObjectsDescend) and full faithfulness (AlgebraicGeometry.Space.DescentSituation.PerfectHomsDescend) — at Situation 09RH's own hypotheses. The only restriction beyond the tag's is#I < B.cutoff.ord.cof, which this formalization is counterexample-forced to carry.The
AlgebraicGeometry.Space.ChartsBoundedbound thatAlgebraicGeometry.Space.DescentSituation.perfectObjectsDescend_and_perfectHomsDescendassumes, and which the source does not state, is discharged here. That statement keeps its own registration:hIimpliesChartsBoundedbut not conversely, so it remains the more general conditional form.
theorem perfectObjectsDescend_and_perfectHomsDescend_of_mk_lt_cof_ord
(hI : Cardinal.mk I < B.cutoff.ord.cof) :
D.PerfectObjectsDescend ∧ D.PerfectHomsDescend
Sources
Section75.25 Cohomology and base change, VI#
Section 75.25 of the Stacks Project, Tag 0A1N. 8 tags here are carried by 7 declarations of the repository (all in StacksFrontier) and 3 registered declarations, quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0A1P, a lemma
Tag 0A1P in the Stacks Project.
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_derivedTensor_and_isIso_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeIso.lean, line 268.
Stacks 0A1P. For
f : X ⟶ Yof finite presentation, a perfectEand a bounded complex𝒢^•of𝒪_X-modules of finite presentation, flat overY, with support proper overY,Rf_*(E ⊗^L 𝒢^•)is perfect, and its formation commutes with arbitrary base change: for every cartesian square,Lg^*Rf_*(E ⊗^L 𝒢^•) ⟶ Rf'_*(L(g')^*E ⊗^L (g')^*𝒢^•)is an isomorphism.
theorem isPerfect_derivedModulePushforward_derivedTensor_and_isIso_baseChange
(f : X ⟶ Y) (hf : IsFinitePresentation B S hS f.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hGb : {i : ℤ | ¬ IsZero (G.X i)}.Finite) (hGfp : ∀ n, (G.X n).IsFinitePresentation)
(hGflat : ∀ n, IsFlatOver B S hS X.2 f.hom (G.X n))
(hGsupp : ∀ n, HasProperSupport B S hS X.2 Y.2 f.hom
(isLocallyOfFiniteType_of_isLocallyOfFinitePresentation B S hS hf.1)
((SheafOfModules.toSheaf _).obj (G.X n))) :
RingedSite.IsPerfect (smallEtaleRingedSite B S hS Y.1 Y.2)
((derivedModulePushforward B S hS f).obj
(((smallEtaleDerivedTensor B S hS X).obj E).obj (DerivedCategory.Q.obj G))) ∧
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedBaseChangeComp B S hS f g f' g' sq.w
(((smallEtaleDerivedTensor B S hS X).obj E).obj (DerivedCategory.Q.obj G))
(((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).obj
(DerivedCategory.Q.obj (((modulePullback B S hS g').mapHomologicalComplex
(ComplexShape.up ℤ)).obj G)))
((smallEtaleDerivedPullbackTensorIso g').hom.app (E, DerivedCategory.Q.obj G) ≫
((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).map
(smallEtaleDerivedModulePullbackQComparison g' G)))
Tag 0A1Q, a remark
Tag 0A1Q in the Stacks Project.
AlgebraicGeometry.Space.exists_finiteProjective_model_derivedSections, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeIso.lean, line 315.
Stacks 0A1Q. Let
Uhave affine underlying schemeSpec R,f : X ⟶ Uof finite presentation and𝒢an𝒪_X-module of finite presentation, flat overU, with support proper overU. There is a bounded complexM^•of finite projectiveR-modules (so thatM^• ⊗ᴸ_R R' = M^• ⊗_R R') and isomorphismsRΓ(X_{R'}, 𝒢_{R'}) ≅ M^• ⊗ᴸ_R R', functorial in theR-algebraR' = Γ(U', 𝒪): compatible with the pullback maps and the composition law of derived extensions of scalars.
theorem exists_finiteProjective_model_derivedSections
{U : BigFppfOver B S hS} [IsAffine ((BigFppfOver.toScheme B S hS).obj U)]
{X : AlgebraicSpaces B S hS} (f : X ⟶ ⟨yoneda.obj U, isAlgebraicSpace_yoneda B S hS U⟩)
(hf : IsFinitePresentation B S hS f.hom)
(G : SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(hGfp : G.IsFinitePresentation) (hGflat : IsFlatOver B S hS X.2 f.hom G)
(hGsupp : HasProperSupport B S hS X.2 (isAlgebraicSpace_yoneda B S hS U) f.hom
(isLocallyOfFiniteType_of_isLocallyOfFinitePresentation B S hS hf.1)
((SheafOfModules.toSheaf _).obj G)) :
∃ (M : CochainComplex (ModuleCat.{u} Γ((BigFppfOver.toScheme B S hS).obj U, ⊤)) ℤ) (a b : ℤ),
M.IsStrictlyGE a ∧ M.IsStrictlyLE b ∧
(∀ i, Module.Finite Γ((BigFppfOver.toScheme B S hS).obj U, ⊤) (M.X i)) ∧
(∀ i, Module.Projective Γ((BigFppfOver.toScheme B S hS).obj U, ⊤) (M.X i)) ∧
(∀ {U' : BigFppfOver B S hS} [IsAffine ((BigFppfOver.toScheme B S hS).obj U')]
(g : U' ⟶ U), IsIso ((representedAffineDerivedExtension B S hS g).comparison.app M)) ∧
∃ φ : ∀ {U' : BigFppfOver B S hS} [IsAffine ((BigFppfOver.toScheme B S hS).obj U')]
(g : U' ⟶ U) {X' : AlgebraicSpaces B S hS}
(f' : X' ⟶ ⟨yoneda.obj U', isAlgebraicSpace_yoneda B S hS U'⟩) (g' : X' ⟶ X),
IsPullback g' f' f (representedSpaceMap B S hS g) →
((representedAffineDerivedSections B S hS U').obj
((derivedModulePushforward B S hS f').obj (DerivedCategory.Q.obj
(((Stacks.Sheaves.RingedToposModules.RingedToposHom.pullbackModules
(X := smallEtaleRingedTopos B S hS _) (Y := smallEtaleRingedTopos B S hS X)
(smallEtaleRingedToposHom B S hS g').toModulesHom).mapHomologicalComplex
(ComplexShape.up ℤ)).obj
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(ComplexShape.up ℤ) 0).obj G)))) ≅
(representedAffineDerivedExtension B S hS g).functor.obj (DerivedCategory.Q.obj M)),
∀ {U' U'' : BigFppfOver B S hS} [IsAffine ((BigFppfOver.toScheme B S hS).obj U')]
[IsAffine ((BigFppfOver.toScheme B S hS).obj U'')] (g : U' ⟶ U) (g₂ : U'' ⟶ U')
{X' X'' : AlgebraicSpaces B S hS}
(f' : X' ⟶ ⟨yoneda.obj U', isAlgebraicSpace_yoneda B S hS U'⟩)
(f'' : X'' ⟶ ⟨yoneda.obj U'', isAlgebraicSpace_yoneda B S hS U''⟩) (g' : X' ⟶ X)
(g₂' : X'' ⟶ X') (sq : IsPullback g' f' f (representedSpaceMap B S hS g))
(sq₂ : IsPullback g₂' f'' f' (representedSpaceMap B S hS g₂))
(sq₃ : IsPullback (g₂' ≫ g') f'' f (representedSpaceMap B S hS (g₂ ≫ g))),
affineBaseDerivedSectionsPullbackMap B S hS g₂ f' f'' g₂' sq₂.w
(((Stacks.Sheaves.RingedToposModules.RingedToposHom.pullbackModules
(X := smallEtaleRingedTopos B S hS _) (Y := smallEtaleRingedTopos B S hS X)
(smallEtaleRingedToposHom B S hS g').toModulesHom).mapHomologicalComplex
(ComplexShape.up ℤ)).obj
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(ComplexShape.up ℤ) 0).obj G)) ≫
(representedAffineDerivedSections B S hS U'').map
((derivedModulePushforward B S hS f'').map (DerivedCategory.Q.map
((NatIso.mapHomologicalComplex (smallEtaleModulePullbackCompIso B S hS g₂' g')
(ComplexShape.up ℤ)).hom.app
((HomologicalComplex.single
(SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X))
(ComplexShape.up ℤ) 0).obj G)))) ≫
(φ (g₂ ≫ g) f'' (g₂' ≫ g') sq₃).hom =
(representedAffineDerivedExtension B S hS g₂).functor.map (φ g f' g' sq).hom ≫
(RingTheory.TorIndependence.totalDerivedExtensionScalarsCompOfEq
(representedAffineCommRingCatHom B S hS g).hom
(representedAffineCommRingCatHom B S hS g₂).hom
(representedAffineCommRingCatHom B S hS (g₂ ≫ g)).hom
(representedAffineRingHom_comp B S hS g₂ g).symm).hom.app
(DerivedCategory.Q.obj M)
Tag 0CTL, a lemma
Tag 0CTL in the Stacks Project.
AlgebraicGeometry.Space.isPseudoCoherent_derivedModulePushforward_derivedTensor_and_isIso_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangePseudoCoherentIso.lean, line 43.
Stacks 0CTL. For
f : X ⟶ Yof finite presentation, a pseudo-coherentEand a bounded above complex𝒢^•of𝒪_X-modules of finite presentation, flat overY, with support proper overY,Rf_*(E ⊗^L 𝒢^•)is pseudo-coherent, and its formation commutes with arbitrary base change: for every cartesian square,Lg^*Rf_*(E ⊗^L 𝒢^•) ⟶ Rf'_*(L(g')^*E ⊗^L (g')^*𝒢^•)is an isomorphism.
theorem isPseudoCoherent_derivedModulePushforward_derivedTensor_and_isIso_baseChange
(f : X ⟶ Y) (hf : IsFinitePresentation B S hS f.hom)
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPseudoCoherent E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hGb : ∃ b : ℤ, ∀ i, b < i → IsZero (G.X i)) (hGfp : ∀ n, (G.X n).IsFinitePresentation)
(hGflat : ∀ n, IsFlatOver B S hS X.2 f.hom (G.X n))
(hGsupp : ∀ n, HasProperSupport B S hS X.2 Y.2 f.hom
(isLocallyOfFiniteType_of_isLocallyOfFinitePresentation B S hS hf.1)
((SheafOfModules.toSheaf _).obj (G.X n))) :
(smallEtaleRingedSite B S hS Y.1 Y.2).IsPseudoCoherent
((derivedModulePushforward B S hS f).obj
(((smallEtaleDerivedTensor B S hS X).obj E).obj (DerivedCategory.Q.obj G))) ∧
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedBaseChangeComp B S hS f g f' g' sq.w
(((smallEtaleDerivedTensor B S hS X).obj E).obj (DerivedCategory.Q.obj G))
(((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).obj
(DerivedCategory.Q.obj (((modulePullback B S hS g').mapHomologicalComplex
(ComplexShape.up ℤ)).obj G)))
((smallEtaleDerivedPullbackTensorIso g').hom.app (E, DerivedCategory.Q.obj G) ≫
((smallEtaleDerivedTensor B S hS X').obj
((smallEtaleDerivedModulePullback B S hS g').obj E)).map
(smallEtaleDerivedModulePullbackQComparison g' G)))
Tag 0CTM, a lemma
Tag 0CTM in the Stacks Project.
Declarations registered for the tag:
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_and_isIso_baseChange_of_isFlat, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeProperTags.lean, line 28 (StacksFrontier): “(1)”.AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_single_and_isIso_baseChange, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeProperTags.lean, line 31 (StacksFrontier): “(2), with 𝒢 flat over Y (the source's 'flat over S' is false)”.
Tag 0CTN, a lemma
Tag 0CTN in the Stacks Project.
AlgebraicGeometry.Space.isPseudoCoherent_derivedModulePushforward_and_isIso_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangePseudoCoherentIso.lean, line 82.
Stacks 0CTN. For a flat and proper morphism
f : X ⟶ Yof finite presentation and a pseudo-coherentE,Rf_*Eis pseudo-coherent, and its formation commutes with arbitrary base change: for every cartesian square, the base change mapLg^*Rf_*E ⟶ Rf'_*L(g')^*Eof Stacks 07A7 is an isomorphism.
theorem isPseudoCoherent_derivedModulePushforward_and_isIso_baseChange
(f : X ⟶ Y) (hfl : IsFlat B S hS f.hom) (hpr : IsProper B S hS f.hom)
(hf : IsFinitePresentation B S hS f.hom)
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPseudoCoherent E) :
(smallEtaleRingedSite B S hS Y.1 Y.2).IsPseudoCoherent
((derivedModulePushforward B S hS f).obj E) ∧
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (g' : X' ⟶ X) (f' : X' ⟶ Y')
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedModuleBaseChangeMap B S hS f g f' g' sq.w E)
Tag 0D3F, a lemma
Tag 0D3F in the Stacks Project.
AlgebraicGeometry.Space.isIso_homEquiv_single_limitComparison_baseRingTensorModule_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ProperFlatLimitComparisonOverIntegers.lean, line 125. The attribute's comment: “Spec(R) is an affine object T of the site, with R = Γ(T, 𝒪_T).”.
Let
f : X ⟶ Tbe proper, flat and of finite presentation, withTaffine andR = Γ(T, 𝒪_T), and let(M_n)be an inverse system ofR-modules with surjective transition maps. Then the canonical map𝒪_X ⊗_R lim M_n ⟶ lim 𝒪_X ⊗_R M_ninduces an isomorphism from the source, an object ofD_QCoh(𝒪_X), toDQ_Xapplied to the target: the morphism𝒪_X ⊗_R lim M_n ⟶ DQ_X(lim 𝒪_X ⊗_R M_n)corresponding to it under the adjunctionD_QCoh(𝒪_X) ⇄ D(𝒪_X)of Stacks 0CR4 (both modules placed in degree zero) is an isomorphism.The source's
Spec(R)is the affine objectTof the site withR = Γ(T, 𝒪_T).
theorem isIso_homEquiv_single_limitComparison_baseRingTensorModule_of_isProper
(f : X ⟶ schemeSpace T) (hf : IsProper B S hS f.hom) (hflat : IsFlat B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom)
(hT : IsAffine ((BigFppfOver.toScheme B S hS).obj T))
(Msys : ℕᵒᵖ ⥤ ModuleCat.{u} (baseRing B S hS T))
(hsurj : ∀ n : ℕ, Function.Surjective (Msys.map (homOfLE (Nat.le_succ n)).op)) :
letI : ∀ (C : Type (u + 1)) [Category.{u + 1} C] [Abelian C], HasDerivedCategory.{u + 1} C :=
fun C _ _ => HasDerivedCategory.standard C
letI := quasicoherentDerivedInclusion_isLeftAdjoint_of_qcqs B S hS B.contains_terminal X.1 X.2
(qcqs_of_isFinitePresentation_schemeSpace B.contains_terminal f hT hfp).1
(qcqs_of_isFinitePresentation_schemeSpace B.contains_terminal f hT hfp).2
IsIso ((Adjunction.ofIsLeftAdjoint (qcohDerivedProp B S hS X).ι).homEquiv
⟨(DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).obj
((baseRingTensorModule B S hS X (baseRingHomOfMap f)).obj (limit Msys)),
qcohDerivedProp_singleFunctor_obj B S hS _
(isQuasicoherentOnSite_baseRingTensorModule B S hS X _ _)⟩
((DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).obj
(limit (Msys ⋙ baseRingTensorModule B S hS X (baseRingHomOfMap f))))
((DerivedCategory.singleFunctor (Modules.{u, u} B S hS X.1 X.2) 0).map
(limit.post Msys (baseRingTensorModule B S hS X (baseRingHomOfMap f)))))
Stacks.Cohomology.SiteModules.isIso_homEquiv_baseRingTensorLimitComparison_iff, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Modules/BaseRingTensorInverseLimitCoherator.lean, line 201. The attribute's comment: “the source's rereading of the headline, proved as an equivalence: the canonical map induces an isomorphism from its source onto DQ_X of its target iff composition with it is bijective on morphisms out of every object of D_QCoh(𝒪_X). Neither side is proved; the bijectivity asserted by the headline under properness, flatness and finite presentation is NOT proved here, and is not assumed. The existence of DQ_X is a named hypothesis, which is Stacks 0CR4 on the small étale site of a quasi-compact quasi-separated algebraic space”.
The two readings of the headline of Stacks 0D3F agree. Let
DQ_Xbe a right adjoint of the inclusionD_QCoh(𝒪_X) ⥤ D(𝒪_X)— on the small étale site of a quasi-compact quasi-separated algebraic space one exists by Stacks 0CR4,AlgebraicGeometry.Space.quasicoherentDerivedInclusion_isLeftAdjoint, and on a general site it is the hypothesisadj— and suppose the source𝒪_X ⊗_R (lim M_n)of the canonical map of Stacks 0D3F lies inD_QCoh(𝒪_X). Then:the canonical map
𝒪_X ⊗_R (lim M_n) ⟶ lim (𝒪_X ⊗_R M_n)induces an isomorphism from its source ontoDQ_Xapplied to its target if and only if composition with it is bijective on morphisms out of every object ofD_QCoh(𝒪_X).This is the source's own rereading of its headline. Neither side is proved: the statement is an equivalence of the two framings, and its proof is
CategoryTheory.Adjunction.bijective_comp_map_iff_isIso_homEquiv, that is the adjunction alone. No properness, flatness or finite-presentation hypothesis appears, and the transition maps of(M_n)are not assumed surjective, because the equivalence needs none of them.
theorem isIso_homEquiv_baseRingTensorLimitComparison_iff
{DQ : DerivedCategory (SheafOfModules.{max u r w} (ringSheafOfSite O)) ⥤
(quasicoherentDerivedOnSite.{u, v, r, w} O).FullSubcategory}
(adj : (quasicoherentDerivedOnSite.{u, v, r, w} O).ι ⊣ DQ)
(hsrc : quasicoherentDerivedOnSite.{u, v, r, w} O
((DerivedCategory.singleFunctor _ 0).obj ((baseRingTensorFunctor φ).obj c.pt))) :
IsIso (adj.homEquiv ⟨_, hsrc⟩ ((DerivedCategory.singleFunctor _ 0).obj d.pt)
((DerivedCategory.singleFunctor _ 0).map
(baseRingTensorLimitComparison φ Msys c d hd))) ↔
∀ E : (quasicoherentDerivedOnSite.{u, v, r, w} O).FullSubcategory,
Function.Bijective
(fun a : E.obj ⟶
(DerivedCategory.singleFunctor _ 0).obj ((baseRingTensorFunctor φ).obj c.pt) =>
a ≫ (DerivedCategory.singleFunctor _ 0).map
(baseRingTensorLimitComparison φ Msys c d hd))
Tag 0CWH, a lemma
Tag 0CWH in the Stacks Project.
Declarations registered for the tag:
AlgebraicGeometry.Space.isPseudoCoherent_representedAffineDerivedSections_tensor, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CountablePseudoCoherentDualityTags.lean, line 36 (StacksFrontier).
Tag 0A1R, a lemma
Tag 0A1R in the Stacks Project.
AlgebraicGeometry.Space.isPerfect_derivedModulePushforward_derivedSheafHom_and_isIso_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeSheafHomIso.lean, line 88.
For
f : X ⟶ Yof finite presentation, a perfectEand a bounded complex𝒢^•of𝒪_X-modules of finite presentation, flat overY, with support proper overY,Rf_*R𝓗om(E, 𝒢^•)is perfect, and its formation commutes with arbitrary base change: for every cartesian square,Lg^*Rf_*R𝓗om(E, 𝒢^•) ⟶ Rf'_*R𝓗om(L(g')^*E, (g')^*𝒢^•)is an isomorphism.
theorem isPerfect_derivedModulePushforward_derivedSheafHom_and_isIso_baseChange
(f : X ⟶ Y) (hf : IsFinitePresentation B S hS f.hom)
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E)
(G : CochainComplex (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) ℤ)
(hGb : {i : ℤ | ¬ IsZero (G.X i)}.Finite) (hGfp : ∀ n, (G.X n).IsFinitePresentation)
(hGflat : ∀ n, IsFlatOver B S hS X.2 f.hom (G.X n))
(hGsupp : ∀ n, HasProperSupport B S hS X.2 Y.2 f.hom
(isLocallyOfFiniteType_of_isLocallyOfFinitePresentation B S hS hf.1)
((SheafOfModules.toSheaf _).obj (G.X n))) :
RingedSite.IsPerfect (smallEtaleRingedSite B S hS Y.1 Y.2)
((derivedModulePushforward B S hS f).obj
(((smallEtaleDerivedSheafHom B S hS X).obj (op E)).obj (DerivedCategory.Q.obj G))) ∧
∀ {X' Y' : AlgebraicSpaces B S hS} (g : Y' ⟶ Y) (f' : X' ⟶ Y') (g' : X' ⟶ X)
(sq : IsPullback g' f' f g),
IsIso (smallEtaleDerivedSheafHomBaseChangeMap f g f' g' sq.w E G)
Sources
Section75.26 Perfect complexes#
Section 75.26 of the Stacks Project, Tag 0D1X. 8 tags here are carried by 20 declarations of the repository (all in StacksFrontier) and 1 registered declaration, quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0D1Y, a lemma
Tag 0D1Y in the Stacks Project.
AlgebraicGeometry.Space.isEtaleLocallyConstructible_setOf_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPoints.lean, line 120. The attribute's comment: “constructible levels, for a Betti-number function given as data”.
Stacks 0D1Y, the level sets of a Betti-number function are étale locally constructible. The Betti-number function is data, characterised by
hβ: the function of Stacks 0D1Y is built, but nothing identifies it with a derived-fibre Betti function on a scheme chart.constructibleLevelsis part (3) of Stacks 0BDI, the same statement for a scheme, which holds for every scheme and is supplied inStacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPointsUnconditional.lean.
theorem isEtaleLocallyConstructible_setOf_eq
(hβ : IsEtaleLocallyDerivedFibreBetti B S hS X β i)
(constructibleLevels : ∀ (Z : Scheme.{u}) (F : DerivedCategory Z.Modules)
(hF : RingedSpace.IsPseudoCoherent (X := Z.toRingedSpace) F) (m : ℕ),
IsLocallyConstructible {z : Z | DerivedFibreBetti.derivedFiberBetti Z F hF i z = m})
(n : ℕ) :
IsEtaleLocallyConstructible B S hS X {x | β x = n}
AlgebraicGeometry.Space.upperSemicontinuous_of_isEtaleLocallyDerivedFibreBetti, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPoints.lean, line 148. The attribute's comment: “upper semicontinuity, for a Betti-number function given as data”.
Stacks 0D1Y, upper semicontinuity of a Betti-number function. The Betti-number function is data, characterised by
hβ: the function of Stacks 0D1Y is built, but nothing identifies it with a derived-fibre Betti function on a scheme chart.upperSemicontinuousBettiis part (2) of Stacks 0BDI, the same statement for a scheme, which holds for every scheme and is supplied inStacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPointsUnconditional.lean.
theorem upperSemicontinuous_of_isEtaleLocallyDerivedFibreBetti
(hX : IsAlgebraicSpace B S hS X)
(hβ : IsEtaleLocallyDerivedFibreBetti B S hS X β i)
(upperSemicontinuousBetti : ∀ (Z : Scheme.{u}) (F : DerivedCategory Z.Modules)
(hF : RingedSpace.IsPseudoCoherent (X := Z.toRingedSpace) F),
UpperSemicontinuous (DerivedFibreBetti.derivedFiberBetti Z F hF i)) :
UpperSemicontinuous β
AlgebraicGeometry.Space.upperSemicontinuous_geometricFibreBettiFunction, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPointsPseudoCoherent.lean, line 215. The attribute's comment: “clause (2)”.
Stacks 0D1Y, clause (2): the functions
β_iare upper semi-continuous. For a pseudo-coherent objectEofD(𝒪_X)on an algebraic spaceXoverS, thei-th Betti-number functionβ_i : |X| → ℕofEis upper semicontinuous.Nothing is assumed beyond pseudo-coherence of
E. The scheme-level input is part (2) of [Stacks, Tag 0BDI].
theorem upperSemicontinuous_geometricFibreBettiFunction
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPseudoCoherent E) (i : ℤ) :
UpperSemicontinuous (geometricFibreBettiFunction B S hS E i)
AlgebraicGeometry.Space.isEtaleLocallyConstructible_setOf_geometricFibreBettiFunction_eq, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BettiNumbersOnPointsPseudoCoherent.lean, line 231. The attribute's comment: “clause (3)”.
Stacks 0D1Y, clause (3): the level sets of
β_iare étale locally constructible. For a pseudo-coherent objectEofD(𝒪_X)on an algebraic spaceXoverSand a natural numbern, the level set{x ∈ |X| : β_i(x) = n}is étale locally constructible in the sense of [Stacks, Tag 0ECU].Nothing is assumed beyond pseudo-coherence of
E. The scheme-level input is part (3) of [Stacks, Tag 0BDI].
theorem isEtaleLocallyConstructible_setOf_geometricFibreBettiFunction_eq
(E : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPseudoCoherent E) (i : ℤ) (n : ℕ) :
IsEtaleLocallyConstructible B S hS X.1 {x | geometricFibreBettiFunction B S hS E i x = n}
AlgebraicGeometry.Space.geometricFibreBettiFunction_derivedPullback, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedFibreBettiFunction.lean, line 179. The attribute's comment: “clause (1)”.
Stacks 0D1Y, clause (1): formation of
β_icommutes with arbitrary base change. For a morphismg : Y ⟶ Xof algebraic spaces the Betti-number function ofLg^*Eis the Betti-number function ofEcomposed with the map|Y| → |X|.
theorem geometricFibreBettiFunction_derivedPullback {Y : AlgebraicSpaces B S hS} (g : Y ⟶ X)
(E : @DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS X)) _ _
(HasDerivedCategory.standard _))
(_hE : @AlgebraicGeometry.RingedSite.IsPseudoCoherent
(smallEtaleRingedSite B S hS X.1 X.2) _ _ _ _ _ _ (HasDerivedCategory.standard _) E)
(i : ℤ) :
geometricFibreBettiFunction B S hS ((smallEtaleDerivedModulePullback B S hS g).obj E) i =
geometricFibreBettiFunction B S hS E i ∘ pointsMap B S hS Y.1 g.hom
Tag 0E0R, a lemma
Tag 0E0R in the Stacks Project.
AlgebraicGeometry.Space.fibreBetti_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/FibreBettiNumbers.lean, line 115. The attribute's comment: “(1)”.
Stacks 0E0R, part (1): the formation of
β_icommutes with arbitrary base change. Forf : X ⟶ Yproper and of finite presentation,𝓕of finite presentation and flat overY, and any morphismg : Y' ⟶ Yof schemes of the site, thei-th Betti-number function of the base change(X ×_Y Y' ⟶ Y', 𝓕_{Y'})isβ_i ∘ g.
theorem fibreBetti_baseChange (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F)
{U' : BigFppfOver B S hS} (g : U' ⟶ U) (i : ℤ) :
fibreBetti (pullback.snd f (representedSpaceMap B S hS g))
((modulePullback B S hS (pullback.fst f (representedSpaceMap B S hS g))).obj F) i =
fibreBetti f F i ∘ ((BigFppfOver.toScheme B S hS).map g).base
AlgebraicGeometry.Space.upperSemicontinuous_fibreBetti, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/FibreBettiNumbers.lean, line 268. The attribute's comment: “(2)”.
Stacks 0E0R, part (2): the Betti numbers of the fibres are upper semicontinuous. For
f : X ⟶ Yproper and of finite presentation and𝓕of finite presentation and flat overY, the functiony ↦ dim_{κ(y)} Hⁱ(X_y, 𝓕_y)is upper semicontinuous onY.
theorem upperSemicontinuous_fibreBetti (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F) (i : ℤ) :
UpperSemicontinuous (fibreBetti f F i)
AlgebraicGeometry.Space.isLocallyConstructible_fibreBetti_preimage, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/FibreBettiNumbers.lean, line 286. The attribute's comment: “(3)”.
Stacks 0E0R, part (3): the level sets of the Betti numbers of the fibres are locally constructible. For
f : X ⟶ Yproper and of finite presentation and𝓕of finite presentation and flat overY, each set{y | dim_{κ(y)} Hⁱ(X_y, 𝓕_y) = n}is locally constructible inY.
theorem isLocallyConstructible_fibreBetti_preimage (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F) (i : ℤ) (n : ℕ) :
Topology.IsLocallyConstructible (fibreBetti f F i ⁻¹' {n})
AlgebraicGeometry.Space.ProperFlatStein.isLocallyConstructible_fibreBetti_preimage, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatStein/FibreBettiLevelSets.lean, line 46. The attribute's comment: “(3)”.
Stacks 0E0R, part (3): the level sets of the Betti numbers of the fibres are locally constructible. For
f : X ⟶ Yproper and of finite presentation and𝓕of finite presentation and flat overY, each set{y | dim_{κ(y)} Hⁱ(X_y, 𝓕_y) = n}is locally constructible inY.
theorem isLocallyConstructible_fibreBetti_preimage (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F) (i : ℤ) (n : ℕ) :
Topology.IsLocallyConstructible (fibreBettiFunction f F i ⁻¹' {n})
AlgebraicGeometry.Space.ProperFlatStein.fibreBetti_baseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatStein/FibreBettiNumbers.lean, line 243. The attribute's comment: “(1)”.
Stacks 0E0R, part (1): the formation of
β_icommutes with arbitrary base change. Forf : X ⟶ Yproper and of finite presentation,𝓕of finite presentation and flat overY, and any morphismg : Y' ⟶ Yof schemes of the site, thei-th Betti-number function of the base change(X ×_Y Y' ⟶ Y', 𝓕_{Y'})isβ_i ∘ g.
theorem fibreBetti_baseChange (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F)
{U' : BigFppfOver B S hS} (g : U' ⟶ U) (i : ℤ) :
fibreBettiFunction (pullback.snd f (representedSpaceMap B S hS g))
((modulePullback B S hS (pullback.fst f (representedSpaceMap B S hS g))).obj F) i =
fibreBettiFunction f F i ∘ ((BigFppfOver.toScheme B S hS).map g).base
AlgebraicGeometry.Space.ProperFlatStein.upperSemicontinuous_fibreBetti, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatStein/FibreBettiNumbers.lean, line 469. The attribute's comment: “(2)”.
Stacks 0E0R, part (2): the Betti numbers of the fibres are upper semicontinuous. For
f : X ⟶ Yproper and of finite presentation and𝓕of finite presentation and flat overY, the functiony ↦ dim_{κ(y)} Hⁱ(X_y, 𝓕_y)is upper semicontinuous onY.
theorem upperSemicontinuous_fibreBetti (hf : IsProper B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom) (F : Modules.{u, u} B S hS X.1 X.2)
(hFfp : F.IsFinitePresentation) (hF : IsFlatOver B S hS X.2 f.hom F) (i : ℤ) :
UpperSemicontinuous (fibreBettiFunction f F i)
Tag 0D1Z, a lemma
Tag 0D1Z in the Stacks Project.
AlgebraicGeometry.Space.isLocallyConstant_derivedFibreEuler, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DerivedFibreEulerLocallyConstant.lean, line 294. References: SGA 6 III (5.5), “(i), special case”.
Stacks 0D1Z. Let
Ybe an algebraic space andKa perfect object ofD(𝒪_Y). The derived-fibre Euler characteristicy ↦ ∑ (-1)^n dim_k H^n(K ⊗^𝐋 k)is locally constant on|Y|.The function is
AlgebraicGeometry.Space.derivedFibreEuler, the alternating sum of the dimensions of the cohomology ofLp^*Kfor a field-valued pointprepresenting the point; byAlgebraicGeometry.Space.derivedFibreEuler_classit may be computed at any representative.
theorem isLocallyConstant_derivedFibreEuler (Y : AlgebraicSpaces B S hS)
(K : DerivedCategory (SheafOfModules.{u} (ringedToposStructureRingSheaf B S hS Y)))
(hK : (smallEtaleRingedSite B S hS Y.1 Y.2).IsPerfect K) :
IsLocallyConstant (derivedFibreEuler Y K)
AlgebraicGeometry.Space.isLocallyConstant_of_isEtaleLocallyDerivedFibreChi, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/EulerCharacteristicOnPoints.lean, line 93. The attribute's comment: “for a derived-fibre Euler characteristic given as data”.
Stacks 0D1Z. The derived-fibre Euler characteristic of a perfect object of
D(𝒪_X)on an algebraic spaceXis locally constant on|X|.The function is data, characterised by
hχ(see the implementation notes); the proof reduces to the case of schemes, Stacks 0BDJ, by étale localization.
theorem isLocallyConstant_of_isEtaleLocallyDerivedFibreChi
(hX : IsAlgebraicSpace B S hS X)
(hχ : IsEtaleLocallyDerivedFibreChi B S hS X χ) :
IsLocallyConstant χ
Tag 0D20, a lemma
Tag 0D20 in the Stacks Project.
AlgebraicGeometry.Space.isFiniteLocallyFreeOfIntRankInDegree_of_isSingleDegreeRankLocus, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SingleDegreeRankOpenSubspace.lean, line 209. The attribute's comment: “clause (1), for an open subspace with the characterising universal property of clause (2); existence of such an open subspace is not proved”.
Clause (1) of Stacks 0D20. On an open subspace with the universal property of clause (2), the object
Eis a finite locally free module of rankrplaced in degreei: the inclusion of the open subspace factors through it by way of the identity, so the universal property applies to it.
theorem isFiniteLocallyFreeOfIntRankInDegree_of_isSingleDegreeRankLocus
(U : OpenSubspaces B S hS X.1)
[(smallEtaleModulePullbackOnHomotopy B S hS
(openSubspaceImmersion B S hS U)).HasLeftDerivedFunctor
(smallEtaleModuleQuasiIso B S hS X)]
(hU : IsSingleDegreeRankLocus B S hS X E i r U) :
IsFiniteLocallyFreeOfIntRankInDegree B S hS (openSubspaceAsSpace B S hS U)
((smallEtaleDerivedModulePullback B S hS (openSubspaceImmersion B S hS U)).obj E) i r
AlgebraicGeometry.Space.eq_of_isSingleDegreeRankLocus, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SingleDegreeRankOpenSubspace.lean, line 241. The attribute's comment: “uniqueness of an open subspace with the characterising universal property of clause (2); existence of such an open subspace is not proved”.
The uniqueness half of "there exists an open subspace characterised by" in Stacks 0D20. At most one open subspace of
Xhas the universal property of clause (2).Each of the two inclusions factors through the other open subspace, because by clause (1) the restriction of
Ealong it has the required shape and the other open subspace's universal property turns that into a factorization; and a subfunctor through which another's inclusion factors contains it.
theorem eq_of_isSingleDegreeRankLocus (U U' : OpenSubspaces B S hS X.1)
[(smallEtaleModulePullbackOnHomotopy B S hS
(openSubspaceImmersion B S hS U)).HasLeftDerivedFunctor
(smallEtaleModuleQuasiIso B S hS X)]
[(smallEtaleModulePullbackOnHomotopy B S hS
(openSubspaceImmersion B S hS U')).HasLeftDerivedFunctor
(smallEtaleModuleQuasiIso B S hS X)]
(hU : IsSingleDegreeRankLocus B S hS X E i r U)
(hU' : IsSingleDegreeRankLocus B S hS X E i r U') :
U = U'
AlgebraicGeometry.Space.exists_isSingleDegreeRankLocus, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/SingleDegreeRankOpenSubspaceExistence.lean, line 103.
Stacks 0D20. Let
Xbe an algebraic space overS, letEbe a perfect object ofD(𝒪_X)and leti, r ∈ ℤ. There is an open subspaceU ⊆ X, and only one, characterised by: (1)E|_U ≅ M[-i]for a finite locally free𝒪_U-moduleMof rankr(soM = H^i(E|_U)); (2) a morphismf : Y → Xfactors throughUif and only ifLf^*Eis a finite locally free module of rankrplaced in degreei.
theorem exists_isSingleDegreeRankLocus
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E) :
∃ U : OpenSubspaces B S hS X.1,
IsFiniteLocallyFreeOfIntRankInDegree B S hS (openSubspaceAsSpace B S hS U)
((smallEtaleDerivedModulePullback B S hS (openSubspaceImmersion B S hS U)).obj E) i r ∧
IsSingleDegreeRankLocus B S hS X E i r U ∧
∀ U' : OpenSubspaces B S hS X.1, IsSingleDegreeRankLocus B S hS X E i r U' → U' = U
Tag 0E6A, a lemma
Tag 0E6A in the Stacks Project.
Declarations registered for the tag:
AlgebraicGeometry.Space.exists_openSubspace_iff_isFiniteLocallyFreeOfIntRankInDegree_of_isProper, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CohomologyBaseChangeProperTags.lean, line 34 (StacksFrontier).
Tag 0D21, a lemma
Tag 0D21 in the Stacks Project.
AlgebraicGeometry.Space.BottomRankLocus.exists_isBottomRankStratum_isFinitePresentation, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BottomRankStratumExistence.lean, line 582.
Stacks 0D21. Let
Ebe a perfect object ofD(𝒪_X)of Tor amplitude in[a, b]andr ≥ 0. There is an immersionj : Z ⟶ Xof algebraic spaces, of finite presentation, such that (1)H^a(Lj^*E)is finite locally free of rankr, (2) a morphismf : Y ⟶ Xfactors throughjexactly whenH^a(L(f ∘ g)^*E)is finite locally free of rankrfor everyg : Y' ⟶ Y, (3) forf = j ∘ gwe haveH^a(Lf^*E) ≅ g^*H^a(Lj^*E), and (4) ifβ_a(p) ≤ rat every geometric pointpofX(bettiAt, the dimension ofH^a(E ⊗^L k(p))), thenjis a closed immersion,ffactors throughjexactly whenH^a(Lf^*E)is finite locally free of rankr, and, forr = 1, exactly when𝒪_Y → ℋom(H^a(Lf^*E), H^a(Lf^*E))is injective.The source writes
H^0in (4)(b) and (4)(c); fora ≠ 0that reading is false: forE = 𝒪_X[-a]andr = 1the stratum is all ofX, so everyf : Y ⟶ Xfactors through it, whileH^0(Lf^*E) = 0is not locally free of rank1whenYis nonempty. The statement here usesH^a, as in the corrected scheme version [stacks 0BDL].
theorem exists_isBottomRankStratum_isFinitePresentation
(T : DerivedTensorProduct (smallEtaleRingedSite B S hS X.1 X.2))
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E) (b : ℤ)
(hT : DerivedTensorProduct.HasTorAmplitude T E a b) (r : ℕ) :
∃ (Z : AlgebraicSpaces B S hS) (j : Z ⟶ X),
IsBottomRankStratum E a r j ∧ IsFinitePresentation B S hS j.hom ∧
(∀ (Y : AlgebraicSpaces B S hS) (g : Y ⟶ Z),
Nonempty (pullbackHomology E a (g ≫ j) ≅ (pb g).obj (pullbackHomology E a j))) ∧
((∀ p : GeometricPoint B S hS X.1, bettiAt E a p.val.siteObject p.val.value ≤ r) →
IsClosedImmersion B S hS j.hom ∧ (∀ (Y : AlgebraicSpaces B S hS) (f : Y ⟶ X),
((∃ g : Y ⟶ Z, g ≫ j = f) ↔
SheafOfModules.IsFiniteLocallyFreeOfIntRank (pullbackHomology E a f) (r : ℤ))) ∧
(r = 1 → ∀ (Y : AlgebraicSpaces B S hS) (f : Y ⟶ X),
((∃ g : Y ⟶ Z, g ≫ j = f) ↔
AlgebraicGeometry.SheafOfModules.IsScalarFaithful (pullbackHomology E a f))))
Tag 0E6B, a lemma
Tag 0E6B in the Stacks Project.
AlgebraicGeometry.Space.exists_presieve_isPerfect_hasTorAmplitudeGE_iso_etaleRestrict, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatDerivedPushforwardFibreSurjective.lean, line 457. The attribute's comment: “(b)”.
Stacks 0E6B, part (b). Let
f : X ⟶ Ybe proper, flat and of finite presentation, such thatk = H⁰(X_k, 𝒪_{X_k})for every morphismSpec k ⟶ Yfrom the spectrum of a field. Then étale locally onYwe haveRf_*𝒪_X ≅ 𝒪_Y ⊕ PinD(𝒪_Y)withPperfect of Tor amplitude in[1, ∞): every étale chartUofYhas a covering family of chartsVover which the restrictions satisfy(Rf_*𝒪_X)|_V ≅ 𝒪_Y|_V ⊕ P.
theorem exists_presieve_isPerfect_hasTorAmplitudeGE_iso_etaleRestrict (f : X ⟶ Y)
(hf : IsProper B S hS f.hom) (hfl : IsFlat B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom)
(h : ∀ p : FieldValuedPoint B S hS Y.1, Function.Bijective (algebraMap p.carrier
(globalFunctions B (Spec (.of p.carrier)) p.contained
(fibreOverFieldValuedPoint B S hS f.hom p))))
(U : SmallEtale B S hS Y.1) :
∃ R : Presieve U, Sieve.generate R ∈ SmallEtale.topology B S hS Y.1 U ∧
∀ ⦃V : SmallEtale B S hS Y.1⦄ (g : V ⟶ U), R g →
∃ P : DerivedCategory (Modules.{u, u} B S hS (chartSpace B S hS Y V).1
(chartSpace B S hS Y V).2),
(smallEtaleRingedSite B S hS (chartSpace B S hS Y V).1
(chartSpace B S hS Y V).2).IsPerfect P ∧
Stacks.SheafOfModules.DerivedTensorProduct.HasTorAmplitudeGE
(smallEtaleDerivedTensorProduct B S hS (chartSpace B S hS Y V)) P 1 ∧
Nonempty ((etaleModulePullback.{u, u} B S hS (chartToSpace B S hS Y V)
(isEtale_chartToSpace B S hS Y V)).mapDerivedCategory.obj
((derivedModulePushforward B S hS f).obj
((DerivedCategory.singleFunctor _ 0).obj (structureModule X))) ≅
(etaleModulePullback.{u, u} B S hS (chartToSpace B S hS Y V)
(isEtale_chartToSpace B S hS Y V)).mapDerivedCategory.obj
((DerivedCategory.singleFunctor _ 0).obj (structureModule Y)) ⊞ P)
AlgebraicGeometry.Space.isIso_structureSharp_and_baseChange_of_bijective_algebraMap_fibre, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatStructureSheafPushforward.lean, line 73. The attribute's comment: “(a)”.
Let
f : X ⟶ Ybe a morphism of algebraic spaces which is proper, flat and of finite presentation, such thatk = H⁰(X_k, 𝒪_{X_k})for every morphismSpec k ⟶ Ywithka field. Thenf_*𝒪_X = 𝒪_Y— the structure map𝒪_Y ⟶ f_*𝒪_Xis an isomorphism — and this holds after any base changeY' ⟶ Y.
theorem isIso_structureSharp_and_baseChange_of_bijective_algebraMap_fibre (f : X ⟶ Y)
(hf : IsProper B S hS f.hom) (hfl : IsFlat B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom)
(h : ∀ p : FieldValuedPoint B S hS Y.1, Function.Bijective (algebraMap p.carrier
(globalFunctions B (Spec (.of p.carrier)) p.contained
(fibreOverFieldValuedPoint B S hS f.hom p)))) :
IsIso (structureSharp B S hS f) ∧
∀ (Y' : AlgebraicSpaces B S hS) (g : Y' ⟶ Y),
IsIso (structureSharp B S hS (pullback.snd f g))
Tag 0E0S, a lemma
Tag 0E0S in the Stacks Project.
AlgebraicGeometry.Space.isIso_structureSharp_and_baseChange_of_geometricFibres, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/ProperFlatStructureSheafPushforward.lean, line 103.
Let
f : X ⟶ Ybe a morphism of algebraic spaces which is proper, flat and of finite presentation, and whose geometric fibres are reduced and connected. Thenf_*𝒪_X = 𝒪_Y— the structure map𝒪_Y ⟶ f_*𝒪_Xis an isomorphism — and this holds after any base changeY' ⟶ Y.
theorem isIso_structureSharp_and_baseChange_of_geometricFibres (f : X ⟶ Y)
(hf : IsProper B S hS f.hom) (hfl : IsFlat B S hS f.hom)
(hfp : IsFinitePresentation B S hS f.hom)
(hred : ∀ y : GeometricPoint B S hS Y.1,
HasEtaleLocalProperty B (Spec (.of y.1.carrier)) y.1.contained
(fibreOverFieldValuedPoint B S hS f.hom y.1)
(fun T : Scheme.{u} => _root_.AlgebraicGeometry.IsReduced T))
(hconn : ∀ y : GeometricPoint B S hS Y.1,
ConnectedPoints B (Spec (.of y.1.carrier)) y.1.contained
(fibreOverFieldValuedPoint B S hS f.hom y.1)) :
IsIso (structureSharp B S hS f) ∧
∀ (Y' : AlgebraicSpaces B S hS) (g : Y' ⟶ Y),
IsIso (structureSharp B S hS (pullback.snd f g))
Sources
Section75.27 Other applications#
Section 75.27 of the Stacks Project, Tag 0CRT. 3 tags here are carried by 6 declarations of the repository (all in StacksFrontier) and 1 registered declaration, quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0CRU, a lemma
Tag 0CRU in the Stacks Project.
AlgebraicGeometry.Space.countable_cohomology_derivedTensor_and_ext_of_qcqs_etale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CountableCohomologyAndExtQcqsEtale.lean, line 76.
Stacks 0CRU. Let
Xbe a quasi-compact quasi-separated algebraic space overS, letKbe an object ofD_QCoh(𝒪_X)such that the cohomology sheavesH^q(K)have countable sets of sections over the affine schemes étale overX, and letEbe a perfect object ofD(𝒪_X). Then for every quasi-compact quasi-separated étaleU ⟶ Xand everyi : ℤthe setsH^i(U, K ⊗^𝐋 E)andExt^i(E|_U, K|_U)are countable.Here
Uis an object of the spaces-étale site ofX, and quasi-compactness and quasi-separatedness are those ofUitself; sinceXis quasi-compact and quasi-separated these are equivalent to quasi-compactness and quasi-separatedness of the morphismU ⟶ X.H^i(U, K ⊗^𝐋 E)is thei-th cohomology of the derived global sections overUof the restriction ofK ⊗^𝐋 EtoU, and(-)|_Uis restriction of modules along the étaleU ⟶ X, read on derived categories.The cohomology clause is
AlgebraicGeometry.Space.hasCountableCohomology_derivedTensor_restrict_of_isPerfectat the canonical derived tensor product of the small étale site, and theExtclause isAlgebraicGeometry.Space.countable_ext_of_qcqs_etale_of_countable_sections.
theorem countable_cohomology_derivedTensor_and_ext_of_qcqs_etale
(X : AlgebraicSpaces B S hS) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(K : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
letI := HasDerivedCategory.standard (Modules.{u, u} B S hS X.1 X.2)
(∀ v : SmallEtale B S hS X.1, IsAffineChart B S hS X.1 v → ∀ q : ℤ,
Countable (((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS X.1 X.2) q).obj K.obj).val.obj (op v))) →
∀ E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E →
∀ U : SpacesEtale B S hS X, IsQuasiCompactSpace B S hS U.1.left.1 →
IsAbsolutelyQuasiSeparated B S hS B.contains_terminal U.1.left.1 → ∀ i : ℤ,
letI := HasDerivedCategory.standard (Modules.{u, u} B S hS U.1.left.1 U.1.left.2)
letI := HasDerivedCategory.standard AddCommGrpCat.{u + 1}
Countable ((DerivedCategory.homologyFunctor AddCommGrpCat.{u + 1} i).obj
((derivedGlobalSectionsAb B S hS U.1.left).obj
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj
(((smallEtaleDerivedTensor B S hS X).obj K.obj).obj E)))) ∧
Countable (DerivedCategory.Ext
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj E)
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj K.obj) i)
AlgebraicGeometry.Space.hasCountableCohomology_derivedTensor_restrict_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CountableSectionsDerivedTensorPerfect.lean, line 405. The attribute's comment: “the sets H^i(U, K ⊗^𝐋 E)”.
Let
Xbe a quasi-compact quasi-separated algebraic space overS, letKbe an object ofD_QCoh(𝒪_X)whose cohomology sheaves have countable sets of sections over the affine schemes étale overX, and letEbe a perfect object ofD(𝒪_X). ThenH^i(U, (K ⊗^𝐋_{𝒪_X} E)|_U)is countable for every quasi-compact quasi-separatedUétale overXand everyi.
theorem hasCountableCohomology_derivedTensor_restrict_of_isPerfect
(hZ : B.Contains (⊤_ Scheme.{u})) (hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.1)
(T : Stacks.SheafOfModules.DerivedTensorProduct.{u + 1, u, u, u + 1}
(smallEtaleRingedSite B S hS X.1 X.2))
(K : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hqcK : ∀ q : ℤ, ((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS X.1 X.2) q).obj K).IsQuasicoherentOnSite)
(hcountK : ∀ v : SmallEtale B S hS X.1, IsAffineChart B S hS X.1 v → ∀ q : ℤ,
Countable (((DerivedCategory.homologyFunctor
(Modules.{u, u} B S hS X.1 X.2) q).obj K).val.obj (op v)))
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect.{u + 1, u, u, u, u + 1} E)
(U : SpacesEtale B S hS X) (hUqc : IsQuasiCompactSpace B S hS U.1.left.1)
(hUqs : IsAbsolutelyQuasiSeparated B S hS hZ U.1.left.1) :
HasCountableCohomology ((derivedGlobalSectionsAb B S hS U.1.left).obj
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj
((T.tensor.obj K).obj E)))
AlgebraicGeometry.Space.countable_ext_of_qcqs_etale, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CountableSectionsEtale.lean, line 161. The attribute's comment: “the sets Ext^i(E|_U, K|_U)”.
Stacks 0CRU, the clause on
Ext^i(E|_U, K|_U). LetXbe a quasi-compact quasi-separated algebraic space overS, letKbe an object ofD_QCoh(𝒪_X)and letEbe a perfect object ofD(𝒪_X). Then for every quasi-compact quasi-separated étaleU ⟶ Xthe setExt^i(E|_U, K|_U)is countable.The three hypotheses are the three inputs the source proof takes from elsewhere, and they are the three this library does not supply:
hExtidentifiesExt^i(E|_Y, K|_Y)with thei-th cohomology of a derived-sections objectRΓ Y. This is the content of Stacks 08JJ that the library lacks: the carrier of 08JJ in this repository,Stacks.SheafOfModules.isIso_tensorDualToSheafHom_of_localSummand, proves only the undergived sheaf isomorphismF ⊗ E^∨ ≅ SheafHom(E, F), not the derived identityK ⊗^𝐋 E^∨ ≅ RSheafHom(E, K)that turnsExtinto cohomology.haffis the affine case, i.e. Stacks 0CRP transported to the spaces-étale site. The 0CRP carrier in this repository is about quasi-compact opens of a scheme and assumes its own affine case, so it does not supply this.hMVis the Mayer–Vietoris triangle of an elementary distinguished square, i.e. Stacks 0CRS. The 0CRS carrier here is conditional on part (1) of Stacks 08GW, which is itself unproved.Granted those, the devissage is
hasCountableCohomology_of_qcqs_etale.Perfectness of
Eis part of the statement of Stacks 0CRU and is kept here, but it is not consumed by this proof: in the source argument perfectness is used exactly inside the affine case and inside the reduction ofExtto cohomology, and both of those are the hypotheseshaffandhExt. The binder is therefore written_hE.
theorem countable_ext_of_qcqs_etale (hZ : B.Contains (⊤_ Scheme.{u}))
(hXqc : IsQuasiCompactSpace B S hS X.obj)
(hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.obj)
(K : QuasicoherentDerivedCategory.{u, u} B S hS X.obj X.property)
(E : DerivedCategory (Modules.{u, u} B S hS X.obj X.property))
(_hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.obj X.property) E)
(RΓ : (Y : (BigFppfOver B S hS)ᵒᵖ ⥤ Type u) → (Y ⟶ X.obj) →
DerivedCategory AddCommGrpCat.{u})
(hExt : ∀ (Y : SpacesEtale B S hS X) (hY : IsEtale B S hS Y.obj.hom.hom) (j : ℤ),
Nonempty (DerivedCategory.Ext
((etaleModulePullback.{u, u} B S hS (Y := X) ⟨Y.obj.hom.hom⟩ hY).mapDerivedCategory.obj E)
((etaleModulePullback.{u, u} B S hS (Y := X) ⟨Y.obj.hom.hom⟩ hY).mapDerivedCategory.obj
K.obj) j ≃
(DerivedCategory.homologyFunctor AddCommGrpCat.{u} j).obj
(RΓ Y.obj.left.obj Y.obj.hom.hom)))
(haff : ∀ V : SpacesEtale B S hS X, IsAffineSpace B S hS V.obj.left.obj →
HasCountableCohomology (RΓ V.obj.left.obj V.obj.hom.hom))
(hMV : ∀ (W V : SpacesEtale B S hS X) (U : Subfunctor W.obj.left.obj)
(f : V.obj.left.obj ⟶ W.obj.left.obj), f ≫ W.obj.hom.hom = V.obj.hom.hom →
IsElementaryDistinguishedSquare B S hS W.obj.left.property U f →
∃ (fU : RΓ W.obj.left.obj W.obj.hom.hom ⟶ RΓ U.toFunctor (U.ι ≫ W.obj.hom.hom))
(fV : RΓ W.obj.left.obj W.obj.hom.hom ⟶ RΓ V.obj.left.obj V.obj.hom.hom)
(gU : RΓ U.toFunctor (U.ι ≫ W.obj.hom.hom) ⟶
RΓ (pullback U.ι f) (pullback.fst U.ι f ≫ U.ι ≫ W.obj.hom.hom))
(gV : RΓ V.obj.left.obj V.obj.hom.hom ⟶
RΓ (pullback U.ι f) (pullback.fst U.ι f ≫ U.ι ≫ W.obj.hom.hom))
(δ : RΓ (pullback U.ι f) (pullback.fst U.ι f ≫ U.ι ≫ W.obj.hom.hom) ⟶
(RΓ W.obj.left.obj W.obj.hom.hom)⟦(1 : ℤ)⟧),
Triangle.mk (biprod.lift fU fV) (biprod.desc gU (-gV)) δ ∈
distTriang (DerivedCategory AddCommGrpCat.{u}))
(U : SpacesEtale B S hS X) (hU : IsEtale B S hS U.obj.hom.hom)
(hUqc : IsQuasiCompactSpace B S hS U.obj.left.obj)
(hUqs : IsAbsolutelyQuasiSeparated B S hS hZ U.obj.left.obj) (i : ℤ) :
Countable (DerivedCategory.Ext
((etaleModulePullback.{u, u} B S hS (Y := X) ⟨U.obj.hom.hom⟩ hU).mapDerivedCategory.obj E)
((etaleModulePullback.{u, u} B S hS (Y := X) ⟨U.obj.hom.hom⟩ hU).mapDerivedCategory.obj K.obj)
i)
AlgebraicGeometry.Space.countable_ext_etaleRestrict_of_countable_sections_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ExtRestrictCountableSectionsPerfect.lean, line 152. The attribute's comment: “the sets Ext^i(E|_U, K|_U), at the tag's own hypotheses; the H^i(U, K ⊗^𝐋 E) clause is AlgebraicGeometry.Space.hasCountableCohomology_derivedTensor_restrict_of_isPerfect”.
The
Extclause of [Stacks, Tag 0CRU]. LetXbe a quasi-compact absolutely quasi-separated algebraic space overS, letKbe an object ofD_QCoh(𝒪_X)whose cohomology sheaves have countable sets of sections over the affine schemes étale overX, and letEbe a perfect object ofD(𝒪_X). ThenExt^i_{𝒪_U}(E|_U, K|_U)is countable for every quasi-compact absolutely quasi-separated objectUof the spaces-étale site ofX.Every hypothesis is the tag's own. This is
countable_ext_etaleRestrict_of_countable_sections_derivedDualTensor_of_qcohDerived, in this namespace, with its remaining input — countability of the sections of the cohomology sheaves ofE^∨ ⊗^𝐋 K— supplied byAlgebraicGeometry.Space.countable_sections_homology_derivedDualTensor_of_isPerfect.This is one half of the tag's covering set; the other is the
H^i(U, K ⊗^𝐋 E)clause,AlgebraicGeometry.Space.hasCountableCohomology_derivedTensor_restrict_of_isPerfect.
theorem countable_ext_etaleRestrict_of_countable_sections_of_isPerfect
(X : AlgebraicSpaces B S hS) (hZ : B.Contains (⊤_ Scheme.{u}))
(hXqc : IsQuasiCompactSpace B S hS X.1)
(hXqs : IsAbsolutelyQuasiSeparated B S hS hZ X.1)
(K : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hqcK : ∀ q : ℤ, ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) q).obj
K).IsQuasicoherentOnSite)
(hcountK : ∀ v : SmallEtale B S hS X.1, IsAffineChart B S hS X.1 v → ∀ q : ℤ,
Countable (((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X.1 X.2) q).obj
K).val.obj (op v)))
(E : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hE : RingedSite.IsPerfect (smallEtaleRingedSite B S hS X.1 X.2) E)
(U : SpacesEtale B S hS X) (hUqc : IsQuasiCompactSpace B S hS U.1.left.1)
(hUqs : IsAbsolutelyQuasiSeparated B S hS hZ U.1.left.1) (i : ℤ) :
Countable (DerivedCategory.Ext
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj E)
((etaleModulePullback.{u, u} B S hS U.1.hom U.2).mapDerivedCategory.obj K) i)
Tag 0CRV, a lemma
Tag 0CRV in the Stacks Project.
Declarations registered for the tag:
AlgebraicGeometry.Space.exists_isHomotopyColimit_isPerfect_iff_countable, declared in the repository, registered withadd_to_stacksin StacksFrontier/AlgebraicGeometry/Spaces/Perfect/CountablePseudoCoherentDualityTags.lean, line 34 (StacksFrontier).
Tag 0CRW, a lemma
Tag 0CRW in the Stacks Project.
AlgebraicGeometry.Space.exists_isHomotopyColimitCocone_derivedSections_tensor_of_isFlat, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/FlatPullbackSectionsHomotopyColimit.lean, line 77. The attribute's comment: “part (2)”.
Stacks 0CRW (2). Let
Abe a ring andf : U ⟶ Xa flat morphism of algebraic spaces of finite presentation overA. There is a system(Eₙ)of perfect objects ofD(𝒪_X)withRΓ(U, Lf^*K) = hocolim RΓ(X, Eₙ ⊗^L K)inD(A)functorially inK ∈ D_QCoh(𝒪_X): natural mapsRΓ(X, Eₙ ⊗^L -) ⟶ RΓ(U, Lf^* -)exhibit the homotopy colimit at every suchK.
theorem exists_isHomotopyColimitCocone_derivedSections_tensor_of_isFlat
{U X : AlgebraicSpaces B (Spec (.of A)) hA} (f : U ⟶ X)
(hX : IsFinitePresentation B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X).hom)
(hU : IsFinitePresentation B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA U).hom)
(hf : IsFlat B (Spec (.of A)) hA f.hom) :
∃ E : ℕ ⥤ DerivedCategory (Modules.{u, u} B (Spec (.of A)) hA X.1 X.2),
(∀ n, (smallEtaleRingedSite B (Spec (.of A)) hA X.1 X.2).IsPerfect (E.obj n)) ∧
∃ ι : ∀ n, (smallEtaleDerivedTensor B (Spec (.of A)) hA X).obj (E.obj n) ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA) ⟶
smallEtaleDerivedModulePullback B (Spec (.of A)) hA f ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA U) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA),
∀ K, QcohDerived B (Spec (.of A)) hA X K →
IsHomotopyColimitCocone
(E ⋙ (smallEtaleDerivedTensor B (Spec (.of A)) hA X).flip.obj K ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA))
fun n ↦ (ι n).app K
AlgebraicGeometry.Space.exists_isHomotopyColimitCocone_derivedHom_of_isFlat, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/FlatPullbackSectionsHomotopyColimit.lean, line 114. The attribute's comment: “part (1)”.
Stacks 0CRW (1). Let
Abe a ring andf : U ⟶ Xa flat morphism of algebraic spaces of finite presentation overA. There is an inverse system(Lₙ)of perfect objects ofD(𝒪_X)withRΓ(U, Lf^*K) = hocolim RHom_X(Lₙ, K)inD(A)functorially inK ∈ D_QCoh(𝒪_X), whereRHom_X(L, K) = RΓ(X, R𝓗om(L, K)): natural mapsRHom_X(Lₙ, -) ⟶ RΓ(U, Lf^* -)exhibit the homotopy colimit at every suchK.
theorem exists_isHomotopyColimitCocone_derivedHom_of_isFlat
{U X : AlgebraicSpaces B (Spec (.of A)) hA} (f : U ⟶ X)
(hX : IsFinitePresentation B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X).hom)
(hU : IsFinitePresentation B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA U).hom)
(hf : IsFlat B (Spec (.of A)) hA f.hom) :
∃ L : ℕᵒᵖ ⥤ DerivedCategory (Modules.{u, u} B (Spec (.of A)) hA X.1 X.2),
(∀ n, (smallEtaleRingedSite B (Spec (.of A)) hA X.1 X.2).IsPerfect (L.obj n)) ∧
∃ ι : ∀ n, (smallEtaleDerivedSheafHom B (Spec (.of A)) hA X).obj (op (L.obj (op n))) ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA) ⟶
smallEtaleDerivedModulePullback B (Spec (.of A)) hA f ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA U) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA),
∀ K, QcohDerived B (Spec (.of A)) hA X K →
IsHomotopyColimitCocone
(L.rightOp ⋙ (smallEtaleDerivedSheafHom B (Spec (.of A)) hA X).flip.obj K ⋙
derivedModulePushforward B (Spec (.of A)) hA (toBase B (Spec (.of A)) hA X) ⋙
representedAffineDerivedSections B (Spec (.of A)) hA
(baseSiteObj B (Spec (.of A)) hA))
fun n ↦ (ι n).app K
Sources
Section75.28 The resolution property#
Section 75.28 of the Stacks Project, Tag 0GUR. 4 tags here are carried by 9 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0GUS, a definition
Tag 0GUS in the Stacks Project.
AlgebraicGeometry.Space.HasResolutionProperty, a class in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionProperty.lean, line 55.
Stacks 0GUS. An algebraic space has the resolution property when every quasi-coherent module of finite type on it is a quotient of a finite locally free module.
class HasResolutionProperty (B : BigFppfSite.{u}) (S : Scheme.{u}) (hS : B.Contains S)
(F : (BigFppfOver B S hS)ᵒᵖ ⥤ Type u) (hF : IsAlgebraicSpace B S hS F) : Prop
Tag 0GUT, a lemma
Tag 0GUT in the Stacks Project.
AlgebraicGeometry.Space.exists_affine_etale_cover_hasResolutionProperty_of_isRelativelyAmple, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyRelativeAmple.lean, line 87. The attribute's comment: “the first step of the proof: an affine surjective étale chart of the target over which the scheme fibre product carries an ample module and has the resolution property”.
Let
f : X ⟶ Ybe a morphism of algebraic spaces over a scheme withYquasi-compact, and letLbe anf-ample module onX. Then there is an affine schemeVover the base with a surjective étale morphismgtoY, and a cartesian square presenting the fibre productV ×_Y Xas a schemeU, such that the base change ofLtoUis ample andUhas the resolution property.
theorem exists_affine_etale_cover_hasResolutionProperty_of_isRelativelyAmple
{X Y : AlgebraicSpaces B S hS} {f : X ⟶ Y} {L : Modules.{u, u} B S hS X.1 X.2}
(hL : IsRelativelyAmple B S hS f L) (hY : IsQuasiCompactSpace B S hS Y.1) :
∃ (V : BigFppfOver B S hS) (g : yoneda.obj V ⟶ Y.1),
IsAffine ((BigFppfOver.toScheme B S hS).obj V) ∧
(MorphismProperty.inverseImage
(@Surjective.{u} : MorphismProperty Scheme.{u})
(BigFppfOver.toScheme B S hS)).presheaf g ∧
IsEtale B S hS g ∧
∃ (U : BigFppfOver B S hS) (q : U ⟶ V) (p : yoneda.obj U ⟶ X.1),
IsPullback p (yoneda.map q) f.hom g ∧
(∃ hinv : Stacks.Ample.IsInvertibleModule ((BigFppfOver.toScheme B S hS).obj U)
(baseChangeModule B S hS L U p),
@Stacks.Ample.IsAmple ((BigFppfOver.toScheme B S hS).obj U)
(baseChangeModule B S hS L U p) hinv) ∧
_root_.AlgebraicGeometry.HasResolutionProperty
((BigFppfOver.toScheme B S hS).obj U)
AlgebraicGeometry.Space.hasResolutionProperty_source_of_isRelativelyAmple, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyRelativeAmpleMain.lean, line 163.
Stacks 0GUT. Let
f : X ⟶ Ybe a morphism of algebraic spaces over a schemeSsuch thatYis quasi-compact, quasi-separated and has the resolution property, and such that there is anf-ample invertible𝒪_X-moduleL. ThenXhas the resolution property.This is
AlgebraicGeometry.Space.exists_finiteLocallyFree_epi_of_isRelativelyAmplepackaged as the classAlgebraicGeometry.Space.HasResolutionPropertyof Stacks 0GUS.The hypotheses are exactly the tag's.
hLinvis part of hypothesis (2): Stacks 0D31 defines anf-ample module to be invertible, and in this treeIsRelativelyAmplerecords invertibility only of the base changes ofLto the scheme charts, so it is carried separately.hZis needed only to stateIsAbsolutelyQuasiSeparated, this tree's spelling of quasi-separatedness ofY. Quasi-compactness and quasi-separatedness offare derived fromhLby Stacks 0D34 and 03KK, not assumed.
theorem hasResolutionProperty_source_of_isRelativelyAmple (hZ : B.Contains (⊤_ Scheme.{u}))
{X Y : AlgebraicSpaces B S hS} (f : X ⟶ Y) (L : Modules.{u, u} B S hS X.1 X.2)
(hLinv : IsInvertibleModule X L) (hL : IsRelativelyAmple B S hS f L)
(hYqc : IsQuasiCompactSpace B S hS Y.1) (hYqs : IsAbsolutelyQuasiSeparated B S hS hZ Y.1)
(hYres : HasResolutionProperty.{u, u} B S hS Y.1 Y.2) :
HasResolutionProperty.{u, u} B S hS X.1 X.2
Tag 0GUU, a lemma
Tag 0GUU in the Stacks Project.
AlgebraicGeometry.Space.exists_finiteLocallyFree_modulePullback_hom_of_hasResolutionProperty, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyPushforwardSections.lean, line 151. The attribute's comment: “the step of the proof producing the finite locally free module on the target and its transpose on the source, for a quasi-compact quasi-separated morphism: the sections of the direct image over an affine chart of the target lie in the image submodule”.
The step of the proof of Stacks 0GUU which produces the finite locally free module on the source. In the situation of
AlgebraicGeometry.Space.exists_finiteLocallyFree_epi_submodule_modulePushforwardthe pullbackf^*ℰis again finite locally free, and the mapℰ ⟶ f_*ℱ— the epimorphism onto the submodule𝒢followed by its inclusion — transposes alongf^* ⊣ f_*to a mapψ : f^*ℰ ⟶ ℱ.This is the source's sentence "Then the corresponding maps
f^*ℰ_i ⊗ ℒ^{⊗ n_i} → ℱare going to be jointly surjective" atℒ = 𝒪_X, up to that surjectivity: the module and the map exist, andψis characterized by its transpose beingφ ≫ 𝒢.ι. Finite local freeness off^*ℰis Stacks 03DL under pullback.
theorem exists_finiteLocallyFree_modulePullback_hom_of_hasResolutionProperty
(hZ : B.Contains (⊤_ Scheme.{u})) {X Y : AlgebraicSpaces B S hS} (f : X ⟶ Y)
(hfqc : IsQuasiCompact B S hS f.hom) (hfqs : IsQuasiSeparated B S hS f.hom)
(hYqc : IsQuasiCompactSpace B S hS Y.1)
(hYqs : IsAbsolutelyQuasiSeparated B S hS hZ Y.1)
(hYres : HasResolutionProperty.{u, u} B S hS Y.1 Y.2)
(F : Modules.{u, u} B S hS X.1 X.2) (hF : F.IsQuasicoherentOnSite)
{α : Type*} (s : Finset α) (V : SmallEtale B S hS Y.1) (hV : IsAffine (chartScheme V))
(σ : α → ((modulePushforward B S hS f).obj F).val.obj (op V)) :
∃ (G : ((modulePushforward B S hS f).obj F).Submodule)
(E : Modules.{u, u} B S hS Y.1 Y.2)
(_ : SheafOfModules.IsFiniteLocallyFreeOnSite E)
(φ : E ⟶ G.toSheafOfModules), Epi φ ∧ (∀ x ∈ s, σ x ∈ G.obj (op V)) ∧
SheafOfModules.IsFiniteLocallyFreeOnSite ((modulePullback B S hS f).obj E) ∧
∃ ψ : (modulePullback B S hS f).obj E ⟶ F,
(modulePullbackAdjunction B S hS f).homEquiv E F ψ = φ ≫ G.ι
AlgebraicGeometry.Space.isQuasiAffineHom_of_isAffineHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyQuasiAffine.lean, line 125. The attribute's comment: “the reduction of the hypothesis: the affine case of the statement is its quasi-affine case, an affine morphism of algebraic spaces being quasi-affine”.
The reduction of the affine case of Stacks 0GUU to its quasi-affine case. An affine morphism of algebraic spaces is quasi-affine: its base change to an affine test object is represented by an affine, hence quasi-affine, scheme.
This is why Stacks 0GUU may assume
fquasi-affine throughout: its two hypotheses, affine and quasi-affine, are one hypothesis.
theorem isQuasiAffineHom_of_isAffineHom {X Y : (BigFppfOver B S hS)ᵒᵖ ⥤ Type u} {f : X ⟶ Y}
(hf : IsAffineHom B S hS f) : IsQuasiAffineHom B S hS f
AlgebraicGeometry.Space.exists_affine_etale_cover_hasResolutionProperty_of_isQuasiAffineHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyQuasiAffine.lean, line 168. The attribute's comment: “the chart step of the proof: an affine surjective étale chart of the target over which the scheme fibre product has the resolution property”.
The chart step of the proof of Stacks 0GUU. For
f : X ⟶ Ya quasi-affine morphism of algebraic spaces withYquasi-compact there are an affine schemeVover the base with a surjective étale morphism toYand a cartesian square presenting the fibre productV ×_Y Xas a schemeUwith the resolution property.This is the source's sentence "Choose an affine scheme
Vand a surjective étale morphismV → Y. SetU = V ×_Y X", with the properties ofUthat the rest of the proof reads off it:𝒪_Uis ample onUbecausefis quasi-affine andVis affine (Stacks 0GUQ and Stacks 01VK), soUhas the resolution property by Stacks 0F87.
theorem exists_affine_etale_cover_hasResolutionProperty_of_isQuasiAffineHom
{X Y : AlgebraicSpaces B S hS} (f : X ⟶ Y) (hf : IsQuasiAffineHom B S hS f.hom)
(hY : IsQuasiCompactSpace B S hS Y.1) :
∃ (V : BigFppfOver B S hS) (g : yoneda.obj V ⟶ Y.1),
IsAffine ((BigFppfOver.toScheme B S hS).obj V) ∧
(MorphismProperty.inverseImage
(@Surjective.{u} : MorphismProperty Scheme.{u})
(BigFppfOver.toScheme B S hS)).presheaf g ∧
IsEtale B S hS g ∧
∃ (U : BigFppfOver B S hS) (q : U ⟶ V) (p : yoneda.obj U ⟶ X.1),
IsPullback p (yoneda.map q) f.hom g ∧
_root_.AlgebraicGeometry.HasResolutionProperty
((BigFppfOver.toScheme B S hS).obj U)
AlgebraicGeometry.Space.hasResolutionProperty_source_of_isQuasiAffineHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyQuasiAffineMain.lean, line 86. The attribute's comment: “the quasi-affine case; the affine case is AlgebraicGeometry.Space.hasResolutionProperty_source_of_isAffineHom, and the two together are the tag's "affine or quasi-affine"”.
Stacks 0GUU, the quasi-affine case. For
f : X ⟶ Ya quasi-affine morphism of algebraic spaces overSwithYquasi-compact, quasi-separated and carrying the resolution property,Xhas the resolution property.This is the source's reduction: by Stacks 0GUQ the structure sheaf of
Xisf-ample (AlgebraicGeometry.Space.isRelativelyAmple_unit_of_isQuasiAffineHom) and invertible (AlgebraicGeometry.Space.isInvertibleModule_unit), so this is Stacks 0GUT,AlgebraicGeometry.Space.hasResolutionProperty_source_of_isRelativelyAmple, atL = 𝒪_X.
theorem hasResolutionProperty_source_of_isQuasiAffineHom (hZ : B.Contains (⊤_ Scheme.{u}))
{X Y : AlgebraicSpaces B S hS} (f : X ⟶ Y) (hf : IsQuasiAffineHom B S hS f.hom)
(hYqc : IsQuasiCompactSpace B S hS Y.1) (hYqs : IsAbsolutelyQuasiSeparated B S hS hZ Y.1)
(hYres : HasResolutionProperty.{u, u} B S hS Y.1 Y.2) :
HasResolutionProperty.{u, u} B S hS X.1 X.2
AlgebraicGeometry.Space.hasResolutionProperty_source_of_isAffineHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyQuasiAffineMain.lean, line 103. The attribute's comment: “the affine case; the quasi-affine case is AlgebraicGeometry.Space.hasResolutionProperty_source_of_isQuasiAffineHom, and the two together are the tag's "affine or quasi-affine"”.
Stacks 0GUU, the affine case. For
f : X ⟶ Yan affine morphism of algebraic spaces overSwithYquasi-compact, quasi-separated and carrying the resolution property,Xhas the resolution property.An affine morphism of algebraic spaces is quasi-affine (
AlgebraicGeometry.Space.isQuasiAffineHom_of_isAffineHom), so this is the quasi-affine case.
theorem hasResolutionProperty_source_of_isAffineHom (hZ : B.Contains (⊤_ Scheme.{u}))
{X Y : AlgebraicSpaces B S hS} (f : X ⟶ Y) (hf : IsAffineHom B S hS f.hom)
(hYqc : IsQuasiCompactSpace B S hS Y.1) (hYqs : IsAbsolutelyQuasiSeparated B S hS hZ Y.1)
(hYres : HasResolutionProperty.{u, u} B S hS Y.1 Y.2) :
HasResolutionProperty.{u, u} B S hS X.1 X.2
Tag 0GUV, a lemma
Tag 0GUV in the Stacks Project.
AlgebraicGeometry.Space.hasResolutionProperty_of_isSurjective_isFiniteLocallyFree_unconditional, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/ResolutionPropertyFiniteLocallyFreeTrace.lean, line 185.
Stacks 0GUV, with no further input. For
f : X ⟶ Ya surjective finite locally free morphism of algebraic spaces over a schemeS, ifXhas the resolution property then so doesY.This is
AlgebraicGeometry.Space.hasResolutionProperty_of_isSurjective_isFiniteLocallyFree_of_dualTwistwith its hypothesishdualdischarged byAlgebraicGeometry.Space.directImageDualDescent, which is proved outright for every morphism of algebraic spaces. Its hypotheses are therefore exactly the tag's:fsurjective,ffinite locally free, andXhaving the resolution property.
theorem hasResolutionProperty_of_isSurjective_isFiniteLocallyFree_unconditional (f : X ⟶ Y)
(hsurj : IsSurjective B S hS f.hom) (hflf : IsFiniteLocallyFree B S hS f)
(hX : HasResolutionProperty.{u, u} B S hS X.1 X.2) :
HasResolutionProperty.{u, u} B S hS Y.1 Y.2
Sources
Section75.29 Detecting Boundedness#
Section 75.29 of the Stacks Project, Tag 0GFE. 4 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.
Tag 0GFF, a lemma
Tag 0GFF in the Stacks Project.
AlgebraicGeometry.Space.eventually_subsingleton_hom_iff_truncGE_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DetectingBoundednessTruncGE.lean, line 162.
[Stacks, Tag 0GFF]. Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overS, letPbe a perfect object ofD(𝒪_X), letEbe an object ofD_QCoh(𝒪_X)and letabe an integer. Then
Hom_{D(𝒪_X)}(P⟦-i⟧, E) = 0fori ≫ 0if and only if
Hom_{D(𝒪_X)}(P⟦-i⟧, τ_{≥a}E) = 0fori ≫ 0,where
τ_{≥a}is the canonical truncation ofD(𝒪_X)and vanishing of a Hom-group is subsingletonness of the morphism type, the derived category being additive.Both implications are the truncation triangle
τ_{≤a-1}E ⟶ E ⟶ τ_{≥a}E ⟶ (τ_{≤a-1}E)⟦1⟧: a morphismP⟦-i⟧ ⟶ Ewhich dies inτ_{≥a}Efactors throughτ_{≤a-1}E, and a morphismP⟦-i⟧ ⟶ τ_{≥a}Ewhich dies in(τ_{≤a-1}E)⟦1⟧factors throughE. So each direction needs only thatτ_{≤a-1}Eand(τ_{≤a-1}E)⟦1⟧receive no nonzero morphism fromP⟦-i⟧forilarge; both lie inD_QCoh(𝒪_X)and are bounded above, bya - 1anda - 2, andAlgebraicGeometry.Space.exists_threshold_subsingleton_hom_negShift_of_isPerfectkills both fromi ≥ a - 1 + con, withcthe threshold that perfectness ofPsupplies through [Stacks, Tag 09MB].
theorem eventually_subsingleton_hom_iff_truncGE_of_isPerfect
(hqc : IsQuasiCompactSpace B S hS X) (hqs : IsAbsolutelyQuasiSeparated B S hS hZ X)
[HasDerivedCategory.{w} (Modules.{u, u} B S hS X hX)]
(P E : DerivedCategory (Modules.{u, u} B S hS X hX))
(hP : (smallEtaleRingedSite B S hS X hX).IsPerfect P)
(hE : ∀ j : ℤ, ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X hX) j).obj
E).IsQuasicoherentOnSite)
(a : ℤ) :
(∃ N : ℤ, ∀ i : ℤ, N ≤ i → Subsingleton (P⟦-i⟧ ⟶ E)) ↔
∃ N : ℤ, ∀ i : ℤ, N ≤ i →
Subsingleton (P⟦-i⟧ ⟶ (DerivedCategory.TStructure.t.truncGE a).obj E)
Tag 0GFG, a lemma
Tag 0GFG in the Stacks Project.
AlgebraicGeometry.Space.eventually_subsingleton_hom_iff_truncLE_of_isPerfect, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/DetectingBoundednessTruncLE.lean, line 111.
[Stacks, Tag 0GFG]. Let
Sbe a scheme, letXbe a quasi-compact and quasi-separated algebraic space overS, letPbe a perfect object ofD(𝒪_X), letEbe an object ofD_QCoh(𝒪_X)and letabe an integer. Then
Hom_{D(𝒪_X)}(P⟦-i⟧, E) = 0fori ≪ 0, andHom_{D(𝒪_X)}(P⟦-i⟧, τ_{≤a}E) = 0fori ≪ 0are equivalent.
A perfect object on a quasi-compact algebraic space is bounded above (
AlgebraicGeometry.Space.exists_isLE_of_isPerfect), and for aPbounded above the equivalence is t-structure orthogonality against the truncation triangleτ_{≤a}E ⟶ E ⟶ τ_{≥a+1}E ⟶, which isStacks.DerivedTruncation.eventually_subsingleton_hom_iff_truncLE. Quasi-separatedness ofXand quasi-coherence of the cohomology sheaves ofEare hypotheses of the source that this argument does not read.
theorem eventually_subsingleton_hom_iff_truncLE_of_isPerfect
(hqc : IsQuasiCompactSpace B S hS X)
(_hqs : IsAbsolutelyQuasiSeparated B S hS hZ X)
(P : DerivedCategory (Modules.{u, u} B S hS X hX))
(hP : (smallEtaleRingedSite B S hS X hX).IsPerfect P)
(E : DerivedCategory (Modules.{u, u} B S hS X hX))
(_hE : ∀ i : ℤ, ((DerivedCategory.homologyFunctor (Modules.{u, u} B S hS X hX) i).obj
E).IsQuasicoherentOnSite)
(a : ℤ) :
(∃ N : ℤ, ∀ i : ℤ, i ≤ N → Subsingleton (P⟦-i⟧ ⟶ E)) ↔
∃ N : ℤ, ∀ i : ℤ, i ≤ N →
Subsingleton (P⟦-i⟧ ⟶ (DerivedCategory.TStructure.t.truncLE a).obj E)
Tag 0GFH, a proposition
Tag 0GFH in the Stacks Project.
AlgebraicGeometry.Space.tfae_isLE_of_isPerfect_generator, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BoundedAboveCriterion.lean, line 115.
Stacks 0GFH. Let
Xbe a quasi-compact and quasi-separated algebraic space,Ga perfect object ofD(𝒪_X)which generatesD_QCoh(𝒪_X), andEan object ofD_QCoh(𝒪_X). The following are equivalent: 1.E ∈ D⁻_QCoh(𝒪_X); 2.Hom(G[-i], E) = 0fori ≫ 0; 3.Ext^i_X(G, E) = 0fori ≫ 0; 4.RHom_X(G, E)is bounded above; 5.H^i(X, G^∨ ⊗^L E) = 0fori ≫ 0; 6.RΓ(X, G^∨ ⊗^L E)is bounded above; 7. for every perfectP: (2), (3), (4) hold withPin place ofG,H^i(X, P ⊗^L E) = 0fori ≫ 0, andRΓ(X, P ⊗^L E)is bounded above.
theorem tfae_isLE_of_isPerfect_generator (X : AlgebraicSpaces B S hS)
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(G : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hG : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect G)
(hgen : ∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
(∀ n : ℤ, Subsingleton (G⟦n⟧ ⟶ E.obj)) → IsZero E.obj)
(E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
List.TFAE [
∃ a : ℤ, E.obj.IsLE a,
∃ N : ℤ, ∀ i : ℤ, N ≤ i → Subsingleton (G⟦-i⟧ ⟶ E.obj),
∃ N : ℤ, ∀ i : ℤ, N ≤ i → Subsingleton (G ⟶ E.obj⟦i⟧),
∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedSheafHom B S hS X).obj (op G)).obj E.obj)).IsLE a,
∃ N : ℤ, ∀ i : ℤ, N ≤ i → IsZero ((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2)) i).obj
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj
(smallEtaleDerivedDual B S hS X G)).obj E.obj))),
∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj
(smallEtaleDerivedDual B S hS X G)).obj E.obj)).IsLE a,
∀ P : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P →
(∃ N : ℤ, ∀ i : ℤ, N ≤ i → Subsingleton (P⟦-i⟧ ⟶ E.obj)) ∧
(∃ N : ℤ, ∀ i : ℤ, N ≤ i → Subsingleton (P ⟶ E.obj⟦i⟧)) ∧
(∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedSheafHom B S hS X).obj (op P)).obj E.obj)).IsLE a) ∧
(∃ N : ℤ, ∀ i : ℤ, N ≤ i → IsZero ((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2)) i).obj
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj P).obj E.obj)))) ∧
(∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj P).obj E.obj)).IsLE a)]
Tag 0GFI, a proposition
Tag 0GFI in the Stacks Project.
AlgebraicGeometry.Space.tfae_isGE_of_isPerfect_generator, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Spaces/Perfect/BoundedBelowCriterion.lean, line 133.
Stacks 0GFI. Let
Xbe a quasi-compact and quasi-separated algebraic space,Ga perfect object ofD(𝒪_X)which generatesD_QCoh(𝒪_X), andEan object ofD_QCoh(𝒪_X). The following are equivalent: 1.E ∈ D⁺_QCoh(𝒪_X); 2.Hom(G[-i], E) = 0fori ≪ 0; 3.Ext^i_X(G, E) = 0fori ≪ 0; 4.RHom_X(G, E)is bounded below; 5.H^i(X, G^∨ ⊗^L E) = 0fori ≪ 0; 6.RΓ(X, G^∨ ⊗^L E)is bounded below; 7. for every perfectP: (2), (3), (4) hold withPin place ofG,H^i(X, P ⊗^L E) = 0fori ≪ 0, andRΓ(X, P ⊗^L E)is bounded below.
theorem tfae_isGE_of_isPerfect_generator (X : AlgebraicSpaces B S hS)
(hqc : IsQuasiCompactSpace B S hS X.1)
(hqs : IsAbsolutelyQuasiSeparated B S hS B.contains_terminal X.1)
(G : DerivedCategory (Modules.{u, u} B S hS X.1 X.2))
(hG : (smallEtaleRingedSite B S hS X.1 X.2).IsPerfect G)
(hgen : ∀ E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2,
(∀ n : ℤ, Subsingleton (G⟦n⟧ ⟶ E.obj)) → IsZero E.obj)
(E : QuasicoherentDerivedCategory.{u, u} B S hS X.1 X.2) :
List.TFAE [
∃ a : ℤ, E.obj.IsGE a,
∃ N : ℤ, ∀ i : ℤ, i ≤ N → Subsingleton (G⟦-i⟧ ⟶ E.obj),
∃ N : ℤ, ∀ i : ℤ, i ≤ N → Subsingleton (G ⟶ E.obj⟦i⟧),
∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedSheafHom B S hS X).obj (op G)).obj E.obj)).IsGE a,
∃ N : ℤ, ∀ i : ℤ, i ≤ N → IsZero ((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2)) i).obj
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj
(smallEtaleDerivedDual B S hS X G)).obj E.obj))),
∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj
(smallEtaleDerivedDual B S hS X G)).obj E.obj)).IsGE a,
∀ P : DerivedCategory (Modules.{u, u} B S hS X.1 X.2),
(smallEtaleRingedSite B S hS X.1 X.2).IsPerfect P →
(∃ N : ℤ, ∀ i : ℤ, i ≤ N → Subsingleton (P⟦-i⟧ ⟶ E.obj)) ∧
(∃ N : ℤ, ∀ i : ℤ, i ≤ N → Subsingleton (P ⟶ E.obj⟦i⟧)) ∧
(∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedSheafHom B S hS X).obj (op P)).obj E.obj)).IsGE a) ∧
(∃ N : ℤ, ∀ i : ℤ, i ≤ N → IsZero ((DerivedCategory.homologyFunctor
(ModuleCat.{u + 1} (globalSectionRingSmallEtale B S hS X.1 X.2)) i).obj
((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj P).obj E.obj)))) ∧
(∃ a : ℤ, ((derivedModuleGlobalSectionsSmallEtaleAtModel B S hS X.1 X.2).obj
(((smallEtaleDerivedTensor B S hS X).obj P).obj E.obj)).IsGE a)]
Sources
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