59.86 Preliminaries on base… to 59.98 Comparing chaotic and…

Section59.86 Preliminaries on base change#

Section 59.86 of the Stacks Project, Tag 0EZQ. 6 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 0EZR, a lemma

Tag 0EZR in the Stacks Project.

Stacks.EtaleCohomology.etaleLocalBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/EtaleLocalBaseChangeThreeParts.lean, line 156. The attribute's comment: “parts (1), (2) and (3), as the conjunction of the three implications”.

Stacks 0EZR. Let

X <--h-- Y
|f       |e
S <--g-- T

be a cartesian square of schemes, let {Uᵢ → X} be an étale covering of X such that each composite Uᵢ → X → S factors as Uᵢ → Vᵢ → S with Vᵢ → S étale, and let

Uᵢ <--hᵢ-- Yᵢ
|fᵢ        |eᵢ
Vᵢ <--gᵢ-- Tᵢ

be the induced cartesian squares, with Yᵢ = Uᵢ ×_X Y and Tᵢ = Vᵢ ×_S T. Let F be a sheaf of sets on the small étale site of T, let 𝓖 be an abelian sheaf on it, let K be an object of its unbounded derived category, and write F|ᵢ, 𝓖|ᵢ and K|ᵢ for the inverse images along Tᵢ ⟶ T. Then:

1. if fᵢ^{-1} g_{i,*} F|ᵢ ⟶ h_{i,*} eᵢ^{-1} F|ᵢ is an isomorphism for every i, so is f^{-1} g_* F ⟶ h_* e^{-1} F;
2. if fᵢ^{-1} Rg_{i,*} K|ᵢ ⟶ Rh_{i,*} eᵢ^{-1} K|ᵢ is an isomorphism for every i, so is f^{-1} Rg_* K ⟶ Rh_* e^{-1} K;
3. for every degree q, if fᵢ^{-1} R^q g_{i,*} 𝓖|ᵢ ⟶ R^q h_{i,*} eᵢ^{-1} 𝓖|ᵢ is an isomorphism for every i, so is f^{-1} R^q g_* 𝓖 ⟶ R^q h_* e^{-1} 𝓖.

The hypotheses are exactly the source's: the square is cartesian, the family {Uᵢ → X} is an étale covering, each Uᵢ → S factors through an étale Vᵢ → S, and the two families of induced squares are cartesian. Nothing is assumed of f, g, h, e beyond that, and the coefficients are arbitrary sheaves of sets, arbitrary abelian sheaves and arbitrary objects of the unbounded derived category.

The three parts are proved in Stacks.EtaleCohomology.isIso_smallEtaleSetsBaseChange_app_of_isIso_local, Stacks.EtaleCohomology.isIso_etaleAbelianDerivedBaseChange_app_of_isIso_local and Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_isIso_local; each of them restricts the base change map of the outer square to Uᵢ, compares it there with the base change map of the i-th local square through the two decompositions of the rectangle Uᵢ ⟶ X ⟶ S, and concludes by the locality of invertibility on the small étale site.

theorem etaleLocalBaseChange
    (D : EtaleLocalBaseChangeData.{u, z, _, _} (Type (max (u + 1) v)))
    (𝓖 : Sheaf D.T.smallEtaleTopology Ab.{u + 1})
    (K : DerivedCategory (Sheaf D.T.smallEtaleTopology Ab.{u + 1})) :
    ((∀ i : D.I, IsIso ((smallEtaleSetsBaseChange.{u, v} (D.e_i i) (D.h_i i) (D.g_i i)
          (D.f_i i) (D.local_cartesian i).w.symm).app
            ((smallEtaleInverseImage.{u, v} (D.toT_i i)).obj D.F))) →
        IsIso ((smallEtaleSetsBaseChange.{u, v} D.e D.h D.g D.f
          D.cartesian.w.symm).app D.F)) ∧
      ((∀ i : D.I, IsIso ((etaleAbelianDerivedBaseChange.{u, 0} (D.g_i i) (D.f_i i) (D.e_i i)
            (D.h_i i) (D.local_cartesian i).w.symm).app
              ((etaleAbelianDerivedInverseImage.{u, 0} (D.toT_i i)).obj K))) →
          IsIso ((etaleAbelianDerivedBaseChange.{u, 0} D.g D.f D.e D.h
            D.cartesian.w.symm).app K)) ∧
        ∀ q : ℕ, (∀ i : D.I, IsIso ((higherBaseChangeMapAb (D.f_i i) (D.g_i i) (D.h_i i)
            (D.e_i i) (D.local_cartesian i).w q).app
              ((smallEtalePullbackAb (D.toT_i i)).obj 𝓖))) →
          IsIso ((higherBaseChangeMapAb D.f D.g D.h D.e D.cartesian.w q).app 𝓖)

Stacks.EtaleCohomology.smallEtaleSetsBaseChange, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/OrdinaryLocality.lean, line 283. The attribute's comment: “the set-valued base change map and the locality calculus only: the map g^{-1} f_ to f'_ (g')^{-1} for sheaves of SETS, which the library previously had only for abelian coefficients, together with the hypothesis bundle and the option-free mate calculus the locality argument runs on. NONE of the tag's three clauses is proved here. Clause (1) is blocked at rightSquare_app by a kernel defeq the elaborator cannot reduce because getLimitCone is not exposed by the module system; clause (3) sits on top of it; clause (2) has no proof content anywhere in the repository.”.

The base change map g^{-1} f_* ⟶ f'_* (g')^{-1} for sheaves of sets on the small étale sites of a commutative square g' ≫ f = f' ≫ g of schemes: the mate, under the two inverse-image/direct-image adjunctions, of the identification of the two composite direct images around the square.

public def smallEtaleSetsBaseChange {X' X Y' Y : Scheme.{u}}
    (g' : X' ⟶ X) (f' : X' ⟶ Y') (f : X ⟶ Y) (g : Y' ⟶ Y)
    (w : g' ≫ f = f' ≫ g) :
    smallEtaleDirectImage.{u, v} f ⋙ smallEtaleInverseImage.{u, v} g ⟶
      smallEtaleInverseImage.{u, v} g' ⋙ smallEtaleDirectImage.{u, v} f'

Stacks.EtaleCohomology.isIso_smallEtaleSetsBaseChange_of_isIso_local, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/OrdinaryLocalityCriterion.lean, line 317. The attribute's comment: “clause (1), for sheaves of SETS, which is the generality in which the source states it: the étale-local criterion for the ordinary base change map. Clause (2), the analogue for an object of the unbounded derived category, and clause (3), the analogue for the higher direct images, are not proved here.”.

Stacks 0EZR, part (1): the base change map of sheaves of sets may be checked étale-locally on X. Let

X <--h-- Y
|f       |e
S <--g-- T

be a cartesian square of schemes, let {Uᵢ → X} be an étale covering such that each Uᵢ → S factors as Uᵢ → Vᵢ → S with Vᵢ → S étale, and form the induced cartesian squares with Yᵢ = Uᵢ ×_X Y and Tᵢ = Vᵢ ×_S T. If for every i the base change map fᵢ^{-1} g_{i,*} Fᵢ ⟶ h_{i,*} e_i^{-1} Fᵢ of the i-th local square is an isomorphism, where Fᵢ is the inverse image of F on Tᵢ, then the base change map f^{-1} g_* F ⟶ h_* e^{-1} F of the outer square is an isomorphism.

public theorem isIso_smallEtaleSetsBaseChange_of_isIso_local
    (D : EtaleLocalBaseChangeData.{u, z, max (u + 1) v, (max (u + 1) v) + 1}
      (Type (max (u + 1) v)))
    (hlocal : ∀ i : D.I,
      IsIso ((smallEtaleSetsBaseChange.{u, v} (D.e_i i) (D.h_i i) (D.g_i i) (D.f_i i)
        (D.local_cartesian i).w.symm).app
        ((smallEtaleInverseImage.{u, v} (D.toT_i i)).obj D.F))) :
    IsIso ((smallEtaleSetsBaseChange.{u, v} D.e D.h D.g D.f D.cartesian.w.symm).app D.F)

Tag 0EZS, a lemma

Tag 0EZS in the Stacks Project.

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_paste, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/SmallEtaleHigherBaseChangePasting.lean, line 524. The attribute's comment: “the abelian half: for an abelian sheaf F on T_étale and a degree q, if f⁻¹R^qg_F → R^qh_e⁻¹F and i⁻¹R^qh_e⁻¹F → R^qj_k⁻¹e⁻¹F are isomorphisms then (f∘i)⁻¹R^qg_F → R^qj_(e∘k)⁻¹F is an isomorphism; the derived statement of the same tag is etaleAbelianDerivedBaseChange_isIso_of_comp”.

Base change maps of a rectangle of squares. Consider a tower of two commutative squares of schemes

X₂ --v₂--> X₁ --v₁--> X₀
|g₂         |g₁        |g₀
v           v          v
S₂ --u₂--> S₁ --u₁--> S₀

written in the notation of the Stacks Project by u₁ = f, v₁ = e, g₀ = g, g₁ = h, u₂ = i, v₂ = k, g₂ = j, let F be an abelian sheaf on the small étale site of X₀ and let q be a degree. If the degree-q base change maps f⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F of the lower square and i⁻¹ R^q h_* (e⁻¹ F) ⟶ R^q j_* k⁻¹ (e⁻¹ F) of the upper square are isomorphisms, then so is the degree-q base change map (i ≫ f)⁻¹ R^q g_* F ⟶ R^q j_* (k ≫ e)⁻¹ F of the outer rectangle. Nothing is assumed of the squares beyond commutativity: neither need be cartesian, and the degree q is arbitrary — each degree is a separate statement, with the same degree in the hypotheses and in the conclusion.

public theorem isIso_higherBaseChangeMapAb_app_of_paste (g₀ : X₀ ⟶ S₀) (g₁ : X₁ ⟶ S₁)
    (g₂ : X₂ ⟶ S₂) (u₁ : S₁ ⟶ S₀) (u₂ : S₂ ⟶ S₁) (v₁ : X₁ ⟶ X₀) (v₂ : X₂ ⟶ X₁)
    (w₁ : g₁ ≫ u₁ = v₁ ≫ g₀) (w₂ : g₂ ≫ u₂ = v₂ ≫ g₁) (q : ℕ)
    (F : Sheaf (smallEtaleTopology X₀) Ab.{u + 1})
    (h₁ : IsIso ((higherBaseChangeMapAb u₁ g₀ g₁ v₁ w₁ q).app F))
    (h₂ : IsIso ((higherBaseChangeMapAb u₂ g₁ g₂ v₂ w₂ q).app
      ((smallEtalePullbackAb v₁).obj F))) :
    IsIso ((higherBaseChangeMapAb (u₂ ≫ u₁) g₀ g₂ (v₂ ≫ v₁)
      (smallEtaleRectangle_comm g₀ g₁ g₂ u₁ u₂ v₁ v₂ w₁ w₂) q).app F)

Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange_isIso_of_comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/BaseChangeMate.lean, line 902. The attribute's comment: “the derived half: for K in D(T_étale), if f⁻¹Rg_K → Rh_e⁻¹K and i⁻¹Rh_e⁻¹K → Rj_k⁻¹e⁻¹K are isomorphisms then (f∘i)⁻¹Rg_K → Rj_(e∘k)⁻¹K is an isomorphism; the R^q statement of the same tag is not covered here”.

Consider a tower of two commutative squares of schemes

W <--j-- Z
|i       |k
X <--h-- Y
|f       |e
S <--g-- T

and K in D(T_étale). If the base change maps f⁻¹ Rg_* K ⟶ Rh_* e⁻¹ K and i⁻¹ Rh_* e⁻¹ K ⟶ Rj_* k⁻¹ e⁻¹ K of the two squares are isomorphisms, then so is the base change map (i ≫ f)⁻¹ Rg_* K ⟶ Rj_* (k ≫ e)⁻¹ K of the outer rectangle.

public theorem etaleAbelianDerivedBaseChange_isIso_of_comp (f : X ⟶ S) (g : T ⟶ S) (h : Y ⟶ X)
    (e : Y ⟶ T) (i : W ⟶ X) (j : Z ⟶ W) (k : Z ⟶ Y)
    (sq₁ : e ≫ g = h ≫ f) (sq₂ : k ≫ h = j ≫ i)
    (a : W ⟶ S) (ha : a = i ≫ f) (b : Z ⟶ T) (hb : b = k ≫ e) (hyp : b ≫ g = j ≫ a)
    (K : DerivedCategory (Sheaf T.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}))
    (h₁ : IsIso ((etaleAbelianDerivedBaseChange.{u, v} g f e h sq₁).app K))
    (h₂ : IsIso ((etaleAbelianDerivedBaseChange.{u, v} h i k j sq₂).app
      ((etaleAbelianDerivedInverseImage.{u, v} e).obj K))) :
    IsIso ((etaleAbelianDerivedBaseChange.{u, v} g a b j hyp).app K)

Tag 0EZT, a lemma

Tag 0EZT in the Stacks Project.

Stacks.EtaleCohomology.LimitBaseChange.isPullback_limitSquare, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChange.lean, line 138. The attribute's comment: “the square on the limits of an inverse system of cartesian squares is cartesian”.

The square on the limits is cartesian, the assertion of the setup of Stacks 0EZT. A morphism to X and a morphism to T agreeing over S induce, level by level, a morphism to Y_i through the cartesian square at i; these are compatible with the transitions because the two legs of the square at a larger index determine a morphism, so they define a morphism to Y = lim Y_i, and the same two uniqueness statements make it the unique one.

theorem isPullback_limitSquare (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT)
    (hcY : IsLimit cY)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (heL : ∀ i : Iᵒᵖ, eL ≫ cT.π.app i = cY.π.app i ≫ e.app i)
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i)) :
    IsPullback hL eL fL gL

Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChange.lean, line 306. The attribute's comment: “conclusion (2): base change for the higher direct images passes to the limit”.

Conclusion (2) of Stacks 0EZT. Let (ℱ_i, φ_{i'i}) be a system of abelian sheaves on (T_i) and let ℱ on T be a sheaf whose higher direct images along the limit are presented, as in Stacks 0EYM, as the colimits of the s_i^{-1}R^q g_{i,*}ℱ_i and of the r_i^{-1}R^q h_{i,*}e_i^{-1}ℱ_i, compatibly with the levelwise base change maps. If the base change map of every level is an isomorphism, so is the base change map of the limit square.

theorem isIso_higherBaseChangeMapAb_limit
    (w : ∀ i : Iᵒᵖ, h.app i ≫ f.app i = e.app i ≫ g.app i) (wL : hL ≫ fL = eL ≫ gL)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (𝓕 : Sheaf (smallEtaleTopology cT.pt) Ab.{u + 1})
    (Kg : I ⥤ Sheaf (smallEtaleTopology cS.pt) Ab.{u + 1})
    (ρg : ∀ i : I, Kg.obj i ≅ sourceTerm g cS F q (Opposite.op i))
    (κg : Kg ⟶ (Functor.const I).obj
      (((smallEtalePushforwardAb gL).rightDerived q).obj 𝓕))
    (hκg : IsColimit (Cocone.mk _ κg))
    (Kh : I ⥤ Sheaf (smallEtaleTopology cX.pt) Ab.{u + 1})
    (ρh : ∀ i : I, Kh.obj i ≅ targetTerm h e cX F q (Opposite.op i))
    (κh : Kh ⟶ (Functor.const I).obj
      (((smallEtalePushforwardAb hL).rightDerived q).obj ((smallEtalePullbackAb eL).obj 𝓕)))
    (hκh : IsColimit (Cocone.mk _ κh))
    (α : Kg ⋙ smallEtalePullbackAb fL ⟶ Kh)
    (hα : ∀ i : I, (smallEtalePullbackAb fL).map (ρg i).hom ≫
      stageComparison f g h e cX cS fL F q w hfL (Opposite.op i) = α.app i ≫ (ρh i).hom)
    (hκ : ∀ i : I, (smallEtalePullbackAb fL).map (κg.app i) ≫
      (higherBaseChangeMapAb fL gL hL eL wL q).app 𝓕 = α.app i ≫ κh.app i)
    (hstage : ∀ i : Iᵒᵖ,
      IsIso ((higherBaseChangeMapAb (f.app i) (g.app i) (h.app i) (e.app i) (w i) q).app
        (F.obj i))) :
    IsIso ((higherBaseChangeMapAb fL gL hL eL wL q).app 𝓕)

Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit_of_isLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeColimit.lean, line 525. The attribute's comment: “conclusion (2): if base change for R^q holds at every level of the system of cartesian squares, it holds for the limit square”.

Conclusion (2) of Stacks 0EZT. Let I be a directed set and let (X_i, Y_i, S_i, T_i, f_i, h_i, e_i, g_i) be an inverse system of cartesian squares of schemes with affine transition morphisms and with g_i quasi-compact and quasi-separated, with limit square (X, Y, S, T, f, h, e, g). Let (ℱ_i, φ_{i'i}) be a system of abelian sheaves on (T_i) and ℱ = colim p_i⁻¹ℱ_i. If the base change map f_i⁻¹R^q g_{i, *}ℱ_i ⟶ R^q h_{i, *}e_i⁻¹ℱ_i is an isomorphism for every i, then so is f⁻¹R^q g_*ℱ ⟶ R^q h_*e⁻¹ℱ.

The limit morphisms are any morphisms f, g, h, e satisfying the equations of the projection squares, and ℱ = colim p_i⁻¹ℱ_i is given by maps ι_i : p_i⁻¹ℱ_i ⟶ ℱ compatible with the transition maps and enjoying the universal property of the colimit.

theorem isIso_higherBaseChangeMapAb_limit_of_isLimit [IsDirected I (· ≤ ·)] [Nonempty I]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DX.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DS.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DT.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DY.map a)]
    [∀ i, QuasiCompact (g.app i)] [∀ i, QuasiSeparated (g.app i)]
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i))
    (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT) (hcY : IsLimit cY)
    (gL : cT.pt ⟶ cS.pt) (hL : cY.pt ⟶ cX.pt)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (𝓕 : Sheaf (smallEtaleTopology cT.pt) Ab.{u + 1})
    (ι : ∀ i : Iᵒᵖ, (smallEtalePullbackAb (cT.π.app i)).obj (F.obj i) ⟶ 𝓕)
    (hι : IsPullbackCocone cT F ι) (hpres : IsPullbackColimitPresentation cT 𝓕 F ι)
    (hstage : ∀ i : Iᵒᵖ, IsIso ((higherBaseChangeMapAb (f.app i) (g.app i) (h.app i) (e.app i)
      (hsq i).w q).app (F.obj i))) :
    IsIso ((higherBaseChangeMapAb fL gL hL eL
      (limitSquare_comm f g h e cX cS cT cY fL gL hL eL hcS hfL hgL hhL heL fun i => (hsq i).w)
        q).app 𝓕)

Stacks.EtaleCohomology.LimitSetBaseChange.isIso_smallEtaleSetBaseChange_of_isLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeDirectImageSet.lean, line 588. The attribute's comment: “conclusion (1), at the source's hypotheses: for an inverse system over a directed set of cartesian squares of schemes with affine transition morphisms and quasi-compact quasi-separated g_i, with limit square (f, g, h, e), and a system of sheaves of sets ℱ_i on the T_i (Tag 0EZL) with ℱ = colim p_i^{-1}ℱ_i, if every base change map f_i^{-1}g_{i,*}ℱ_i ⟶ h_{i,*}e_i^{-1}ℱ_i of sheaves of sets (Stacks.EtaleCohomology.smallEtaleSetBaseChange) is an isomorphism then so is f^{-1}g_*ℱ ⟶ h_*e^{-1}ℱ. Together with Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_of_isLimit, conclusion (2), this covers the tag.”.

[Stacks, Tag 0EZT] (1): base change for the direct images of sheaves of sets passes to the limit. Let I be a directed set and let (X_i ← Y_i, S_i ← T_i) be an inverse system over I of cartesian squares of schemes h_i ≫ f_i = e_i ≫ g_i, with affine transition morphisms and with every g_i quasi-compact and quasi-separated. Let X, S, T, Y be the limits, with the limit morphisms f, g, h, e characterised by f ≫ s_i = r_i ≫ f_i, g ≫ s_i = p_i ≫ g_i, h ≫ r_i = q_i ≫ h_i and e ≫ p_i = q_i ≫ e_i. Let (ℱ_i, φ_{i'i}) be a system of sheaves of sets on (T_i) (Stacks 0EZL) and let ℱ = colim p_i^{-1}ℱ_i on T, presented by legs ι_i : p_i^{-1}ℱ_i ⟶ ℱ. If for every i the base change map f_i^{-1}g_{i,*}ℱ_i ⟶ h_{i,*}e_i^{-1}ℱ_i of sheaves of sets is an isomorphism, then so is the base change map f^{-1}g_*ℱ ⟶ h_*e^{-1}ℱ of the limit square.

theorem isIso_smallEtaleSetBaseChange_of_isLimit
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i))
    (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT) (hcY : IsLimit cY)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (heL : ∀ i : Iᵒᵖ, eL ≫ cT.π.app i = cY.π.app i ≫ e.app i)
    (𝓕 : Sheaf (smallEtaleTopology cT.pt) (Type (u + 1)))
    (F : EtaleSheafSystem DT (Type (u + 1)))
    (ι : ∀ i : Iᵒᵖ, (etalePullback (cT.π.app i) (Type (u + 1))).obj (F.obj i) ⟶ 𝓕)
    (hι : ∀ ⦃i j : Iᵒᵖ⦄ (a : i ⟶ j),
      (Adjunction.leftAdjointCompIso (systemAdjunction DT (Type (u + 1)) a)
          (etaleAdjunction (cT.π.app i) (Type (u + 1)))
          (etaleAdjunction (cT.π.app j) (Type (u + 1)))
          (coneProjectionComp (Type (u + 1)) cT a)).inv.app (F.obj j) ≫
        (etalePullback (cT.π.app i) (Type (u + 1))).map (F.map a) ≫ ι i = ι j)
    (hpres : IsSetPullbackColimitPresentation cT F 𝓕 ι)
    (hstage : ∀ i : Iᵒᵖ,
      IsIso ((smallEtaleSetBaseChange.{u, 0} (e.app i) (h.app i) (g.app i) (f.app i)
        (hsq i).w.symm).app (F.obj i))) :
    IsIso ((smallEtaleSetBaseChange.{u, 0} eL hL gL fL
      (LimitBaseChange.limitSquare_comm f g h e cX cS cT cY fL gL hL eL hcS hfL hgL hhL heL
        fun i => (hsq i).w).symm).app 𝓕)

Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_of_isLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeHigherDirectImage.lean, line 339. The attribute's comment: “conclusion (2), at the source's hypotheses: for an inverse system over a directed set of cartesian squares of schemes with affine transition morphisms and quasi-compact quasi-separated g_i, with limit square (f, g, h, e), and a system of abelian sheaves ℱ_i on the T_i with ℱ = colim p_i^{-1}ℱ_i, if every base change map f_i^{-1}R^q g_{i,*}ℱ_i ⟶ R^q h_{i,*}e_i^{-1}ℱ_i is an isomorphism then so is f^{-1}R^q g_*ℱ ⟶ R^q h_*e^{-1}ℱ. Together with Stacks.EtaleCohomology.LimitSetBaseChange.isIso_smallEtaleSetBaseChange_of_isLimit, conclusion (1), this covers the tag.”.

[Stacks, Tag 0EZT] (2): base change for the higher direct images passes to the limit. Let I be a directed set and let (X_i ← Y_i, S_i ← T_i) be an inverse system over I of cartesian squares of schemes h_i ≫ f_i = e_i ≫ g_i, with affine transition morphisms and with every g_i quasi-compact and quasi-separated. Let X, S, T, Y be the limits, with the limit morphisms f, g, h, e characterised by f ≫ s_i = r_i ≫ f_i, g ≫ s_i = p_i ≫ g_i, h ≫ r_i = q_i ≫ h_i and e ≫ p_i = q_i ≫ e_i. Let (ℱ_i, φ_{i'i}) be a system of abelian sheaves on (T_i) (Stacks 0EZL) and let ℱ = colim p_i^{-1}ℱ_i on T, presented by legs ι_i : p_i^{-1}ℱ_i ⟶ ℱ. If for every i the base change map f_i^{-1}R^q g_{i,*}ℱ_i ⟶ R^q h_{i,*}e_i^{-1}ℱ_i is an isomorphism, then so is the base change map f^{-1}R^q g_*ℱ ⟶ R^q h_*e^{-1}ℱ of the limit square.

theorem isIso_higherBaseChangeMapAb_of_isLimit
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i))
    (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT) (hcY : IsLimit cY)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (heL : ∀ i : Iᵒᵖ, eL ≫ cT.π.app i = cY.π.app i ≫ e.app i) (q : ℕ)
    (𝓕 : Sheaf (smallEtaleTopology cT.pt) Ab.{u + 1})
    (F : EtaleSheafSystem DT Ab.{u + 1})
    (ι : ∀ i : Iᵒᵖ, (smallEtalePullbackAb (cT.π.app i)).obj (F.obj i) ⟶ 𝓕)
    (hι : ∀ {i j : Iᵒᵖ} (a : i ⟶ j),
      (conePullbackComp cT a).inv.app (F.obj j) ≫
        (smallEtalePullbackAb (cT.π.app i)).map (F.map a) ≫ ι i = ι j)
    (hpres : IsPullbackColimitPresentation cT 𝓕 F ι)
    (hstage : ∀ i : Iᵒᵖ,
      IsIso ((higherBaseChangeMapAb (f.app i) (g.app i) (h.app i) (e.app i) (hsq i).w q).app
        (F.obj i))) :
    IsIso ((higherBaseChangeMapAb fL gL hL eL
      (limitSquare_comm f g h e cX cS cT cY fL gL hL eL hcS hfL hgL hhL heL fun i => (hsq i).w)
      q).app 𝓕)

Stacks.EtaleCohomology.LimitBaseChange.isIso_smallEtaleSheafBaseChange_limit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeSetsColimit.lean, line 408. The attribute's comment: “conclusion (1): if base change for sheaves of sets holds at every level of the system of cartesian squares, it holds for the limit square”.

Conclusion (1) of Stacks 0EZT. Let I be a directed set and let (X_i, Y_i, S_i, T_i, f_i, h_i, e_i, g_i) be an inverse system of cartesian squares of schemes with affine transition morphisms and with g_i quasi-compact and quasi-separated, with limit square (X, Y, S, T, f, h, e, g). Let (ℱ_i, φ_{i'i}) be a system of sheaves of sets on (T_i) and ℱ = colim p_i⁻¹ℱ_i. If the base change map f_i⁻¹g_{i, *}ℱ_i ⟶ h_{i, *}e_i⁻¹ℱ_i is an isomorphism for every i, then so is f⁻¹g_*ℱ ⟶ h_*e⁻¹ℱ.

The limit morphisms are any morphisms f, g, h, e satisfying the equations of the projection squares, and ℱ = colim p_i⁻¹ℱ_i is given by maps ι_i : p_i⁻¹ℱ_i ⟶ ℱ compatible with the transition maps and enjoying the universal property of the colimit.

theorem isIso_smallEtaleSheafBaseChange_limit {I : Type u} [Preorder I] [IsDirected I (· ≤ ·)]
    [Nonempty I] {DX DS DT DY : Iᵒᵖ ⥤ Scheme.{u}}
    (f : DX ⟶ DS) (g : DT ⟶ DS) (h : DY ⟶ DX) (e : DY ⟶ DT)
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DX.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DS.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DT.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DY.map a)]
    [∀ i, QuasiCompact (g.app i)] [∀ i, QuasiSeparated (g.app i)]
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i))
    {cX : Cone DX} {cS : Cone DS} {cT : Cone DT} {cY : Cone DY}
    (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT) (hcY : IsLimit cY)
    (fL : cX.pt ⟶ cS.pt) (gL : cT.pt ⟶ cS.pt) (hL : cY.pt ⟶ cX.pt) (eL : cY.pt ⟶ cT.pt)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (heL : ∀ i : Iᵒᵖ, eL ≫ cT.π.app i = cY.π.app i ≫ e.app i)
    (F : EtaleSheafSystem DT (Type (u + 1))) (𝓕 : Sheaf (smallEtaleTopology cT.pt) (Type (u + 1)))
    (ι : ∀ i : Iᵒᵖ, (smallEtaleSheafPullback (Type (u + 1)) (X := cT.pt) (cT.π.app i)).obj
      (F.obj i) ⟶ 𝓕)
    (hι : F.IsSheafPullbackCocone cT ι) (hpres : F.IsSheafPullbackColimitPresentation cT 𝓕 ι)
    (hstage : ∀ i : Iᵒᵖ, IsIso ((smallEtaleSheafBaseChange (Type (u + 1)) (f.app i) (g.app i)
      (h.app i) (e.app i) (hsq i).w).app (F.obj i))) :
    IsIso ((smallEtaleSheafBaseChange (Type (u + 1)) fL gL hL eL
      (limitSquare_comm f g h e cX cS cT cY fL gL hL eL hcS hfL hgL hhL heL
        fun i => (hsq i).w)).app 𝓕)

Stacks.EtaleCohomology.LimitSetBaseChange.isIso_smallEtaleSetBaseChange_limit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeGeneral.lean, line 287. The attribute's comment: “conclusion (1): base change for the direct images of sheaves of sets passes to the limit. The two colimit presentations g_*ℱ = colim_i s_i^{-1}g_{i,*}ℱ_i and h_*e^{-1}ℱ = colim_i r_i^{-1}h_{i,*}e_i^{-1}ℱ_i are the two instances of Stacks 0EYM that the source's proof cites; this repository does not prove Stacks 0EYM -- Stacks.EtaleCohomology.LimitHigherDirectImage constructs only its comparison maps and says so in its own clause strings -- and the compatibility of the two with the legs is the pasting of base change maps of Stacks 0E47. All three are carried as declared hypotheses (Kg/ρg/κg/hκg, Kh/ρh/κh/hκh, α/hα/hκ), exactly as for conclusion (2) in Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit. What is proved is the source's own argument from those inputs: inverse image is a left adjoint, so it carries the first presentation to a colimit presentation of f^{-1}g_*ℱ, and a map of colimit cocones that is invertible on every term is invertible. Since the two presentations are hypotheses, the proof uses nothing of the inverse systems but the schemes and the morphisms, and the levels are accordingly carried as bare families rather than as functors out of Iᵒᵖ: no functoriality, no limit property of the cones and no cartesianness of the squares is assumed.”.

Conclusion (1) of Stacks 0EZT. Let (ℱ_i) be a family of sheaves of sets on the T_i and let ℱ on T be a sheaf whose direct images along the limit are presented, as in Stacks 0EYM, as the colimits of the s_i^{-1}g_{i,*}ℱ_i and of the r_i^{-1}h_{i,*}e_i^{-1}ℱ_i, compatibly with the levelwise base change maps. If the base change map of every level is an isomorphism, so is the base change map of the limit square.

theorem isIso_smallEtaleSetBaseChange_limit (w : ∀ i, h i ≫ f i = e i ≫ g i)
    (wL : hL ≫ fL = eL ≫ gL) (hfL : ∀ i, fL ≫ s i = r i ≫ f i)
    (𝓕 : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v)))
    (Kg : I ⥤ Sheaf (smallEtaleTopology S) (Type (max (u + 1) v)))
    (ρg : ∀ i, Kg.obj i ≅ sourceTerm.{u, v} g s F i)
    (κg : Kg ⟶ (Functor.const I).obj
      ((smallEtalePushforward gL (Type (max (u + 1) v))).obj 𝓕))
    (hκg : IsColimit (Cocone.mk
      ((smallEtalePushforward gL (Type (max (u + 1) v))).obj 𝓕) κg))
    (Kh : I ⥤ Sheaf (smallEtaleTopology X) (Type (max (u + 1) v)))
    (ρh : ∀ i, Kh.obj i ≅ targetTerm.{u, v} h e r F i)
    (κh : Kh ⟶ (Functor.const I).obj
      ((smallEtalePushforward hL (Type (max (u + 1) v))).obj
        ((smallEtaleInverseImage.{u, v} eL).obj 𝓕)))
    (hκh : IsColimit (Cocone.mk
      ((smallEtalePushforward hL (Type (max (u + 1) v))).obj
        ((smallEtaleInverseImage.{u, v} eL).obj 𝓕)) κh))
    (α : Kg ⋙ smallEtaleInverseImage.{u, v} fL ⟶ Kh)
    (hα : ∀ i, (smallEtaleInverseImage.{u, v} fL).map (ρg i).hom ≫
      stageComparison.{u, v} f g h e r s fL F w hfL i = α.app i ≫ (ρh i).hom)
    (hκ : ∀ i, (smallEtaleInverseImage.{u, v} fL).map (κg.app i) ≫
      setBaseChangeApp.{u, v} eL hL gL fL wL.symm 𝓕 = α.app i ≫ κh.app i)
    (hstage : ∀ i, IsIso ((smallEtaleSetBaseChange.{u, v} (e i) (h i) (g i) (f i)
      (w i).symm).app (F i))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} eL hL gL fL wL.symm).app 𝓕)

Tag 0GJL, a lemma

Tag 0GJL in the Stacks Project.

Stacks.EtaleCohomology.LimitBaseChangeDerived.isIso_smallEtaleDerivedBaseChangeAb_limit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeDerived.lean, line 322. The attribute's comment: “if the derived base change map of the i-th square is an isomorphism at K_i for all i ≥ 0, then the derived base change map of the limit square is an isomorphism at K, with K_i and K the derived inverse images of K_0 ∈ D^+(T_{0, étale}) to T_i and to T. Both steps of the source's proof are carried out: the reduction to cohomology sheaves is isIso_of_forall_isIso_derivedLimitComparison and the colimit argument of Stacks 0EZT, with the reference to Stacks 0EYM replaced by one to Stacks 0GIU, is isIso_derivedLimitComparison. EXTRA HYPOTHESIS: the single bundled hypothesis P grants, in each degree p, the two instances of the colimit assertion of Stacks 0GIU that the source's proof cites — H^p(Rg_*K) = colim_{i ≥ 0} s_i^{-1}H^p(Rg_{i, *}K_i) and H^p(Rh_*Re^{-1}K) = colim_{i ≥ 0} r_i^{-1}H^p(Rh_{i, *}Re_i^{-1}K_i) — presented as in Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit, namely a diagram with terms identified with derivedSourceTerm resp. derivedTargetTerm and a colimit cocone with the stated vertex, together with the comparison of the two diagrams and its compatibility with terms and legs, which is the pasting of base change maps of Stacks 0E47. That colimit assertion is NOT available in this repository: Stacks.EtaleCohomology.LimitHigherDirectImage.pullbackProjectionMap, the carrier of Stacks 0GIU, states in its own clause string that it is not proved and that the colimit cannot even be formed, both because the colimit assertion of Stacks 0EYM is unproved and because the cocycle compatibility of Stacks 0E47 is unavailable for the bounded below derived base change map. The hypothesis does not restate the goal: it says nothing about any base change map being invertible, only that the two sides are the asserted colimits and that the comparison maps assemble.”.

Stacks 0GJL. Let (X_i, Y_i, S_i, T_i, f_i, h_i, e_i, g_i) be an inverse system of squares as in the setup of Stacks 0EZT, with limit square (X, Y, S, T, f, h, e, g). Let 0 ∈ I, let K_0 be an object of D^+(T_{0, étale}), let K_i be the pullback of K_0 to T_i for i ≥ 0 and K the pullback of K_0 to T. If f\_i^\{-1}Rg\_\{i, *}K\_i = Rh\_\{i, *}e\_i^\{-1}K\_i for all i ≥ 0, then f^\{-1}Rg\_*K = Rh\_*e^\{-1}K.

theorem isIso_smallEtaleDerivedBaseChangeAb_limit
    (w : ∀ i : Iᵒᵖ, h.app i ≫ f.app i = e.app i ≫ g.app i) (wL : hL ≫ fL = eL ≫ gL)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    {i₀ : Iᵒᵖ} (K₀ : PlusAb (DT.obj i₀)) (J : Type w₁) [Category.{w₂} J] (σ : J → Iᵒᵖ)
    (τ : ∀ j : J, σ j ⟶ i₀)
    (P : ∀ p : ℤ,
      ColimitPresentation f g h e cX cS cT cY fL gL hL eL w wL hfL K₀ J σ τ p)
    (hstage : ∀ (i : Iᵒᵖ) (a : i ⟶ i₀),
      IsIso ((smallEtaleDerivedBaseChangeAb (f.app i) (g.app i) (h.app i) (e.app i) (w i)).app
        (stageObj K₀ a))) :
    IsIso ((smallEtaleDerivedBaseChangeAb fL gL hL eL wL).app (limitObj cT K₀))

Stacks.EtaleCohomology.LimitBaseChangeDerived.isIso_smallEtaleDerivedBaseChangeAb_limit_of_isLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Colimits/LimitBaseChangeDerivedColimit.lean, line 807.

Let I be a directed set and let (X_i, Y_i, S_i, T_i, f_i, h_i, e_i, g_i) be an inverse system of cartesian squares of schemes with affine transition morphisms and with g_i quasi-compact and quasi-separated, with limit square (X, Y, S, T, f, h, e, g). Let 0 = i₀ ∈ I, let K_0 ∈ D^+(T_{0, étale}), let K_i be the pullback of K_0 to T_i for i ≥ 0 and K the pullback of K_0 to T. If the derived base change map f_i^{-1}Rg_{i, *}K_i ⟶ Rh_{i, *}e_i^{-1}K_i is an isomorphism for all i ≥ 0, then so is f^{-1}Rg_*K ⟶ Rh_*e^{-1}K.

The limit morphisms are any morphisms f, g, h, e satisfying the equations of the projection squares.

theorem isIso_smallEtaleDerivedBaseChangeAb_limit_of_isLimit [IsDirected I (· ≤ ·)] [Nonempty I]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DX.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DS.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DT.map a)]
    [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (DY.map a)]
    [∀ i, QuasiCompact (g.app i)] [∀ i, QuasiSeparated (g.app i)]
    (hsq : ∀ i : Iᵒᵖ, IsPullback (h.app i) (e.app i) (f.app i) (g.app i))
    (hcX : IsLimit cX) (hcS : IsLimit cS) (hcT : IsLimit cT) (hcY : IsLimit cY)
    (fL : cX.pt ⟶ cS.pt) (gL : cT.pt ⟶ cS.pt) (hL : cY.pt ⟶ cX.pt) (eL : cY.pt ⟶ cT.pt)
    (hfL : ∀ i : Iᵒᵖ, fL ≫ cS.π.app i = cX.π.app i ≫ f.app i)
    (hgL : ∀ i : Iᵒᵖ, gL ≫ cS.π.app i = cT.π.app i ≫ g.app i)
    (hhL : ∀ i : Iᵒᵖ, hL ≫ cX.π.app i = cY.π.app i ≫ h.app i)
    (heL : ∀ i : Iᵒᵖ, eL ≫ cT.π.app i = cY.π.app i ≫ e.app i)
    (i₀ : I) (K₀ : PlusAb (DT.obj (op i₀)))
    (hstage : ∀ (i : Iᵒᵖ) (a : i ⟶ op i₀),
      IsIso ((smallEtaleDerivedBaseChangeAb (f.app i) (g.app i) (h.app i) (e.app i)
        (hsq i).w).app (stageObj K₀ a))) :
    IsIso ((smallEtaleDerivedBaseChangeAb fL gL hL eL
      (LimitBaseChange.limitSquare_comm f g h e cX cS cT cY fL gL hL eL hcS hfL hgL hhL heL
        fun i => (hsq i).w)).app (limitObj cT K₀))

Tag 0EZU, a lemma

Tag 0EZU in the Stacks Project.

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_iff_forall_isIso_stalk, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/StalkwiseCriterion.lean, line 212. The attribute's comment: “base change on higher direct images is an isomorphism iff it is on every stalk”.

Stacks 0EZU. Consider a cartesian diagram of schemes

X  ←-h-  Y
|f       |e
S  ←-g-  T

with g : T ⟶ S quasi-compact and quasi-separated, let F be an abelian sheaf on the small étale site of T, and let q ≥ 0. Then the base change map on higher direct images f⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F is an isomorphism if and only if its map on the stalk at every geometric point of X is an isomorphism.

public theorem isIso_higherBaseChangeMapAb_app_iff_forall_isIso_stalk
    {X Y S T : Scheme.{u}} (f : X ⟶ S) (h : Y ⟶ X) (e : Y ⟶ T) (g : T ⟶ S)
    (sq : IsPullback h e f g) [QuasiCompact g] [QuasiSeparated g]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1}) (q : ℕ) :
    (∀ x : GeometricPoint X,
        IsIso ((smallEtaleStalkAb x.hom).map ((higherBaseChangeMapAb f g h e sq.w q).app F))) ↔
      IsIso ((higherBaseChangeMapAb f g h e sq.w q).app F)

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_iff_forall_isIso_stalkMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/StalkwiseCriterion.lean, line 232. The attribute's comment: “the condition at a geometric point, as a map out of the stalk at its image”.

Stacks 0EZU, with the condition at a geometric point x̄ of X phrased as invertibility of the canonical map (R^q g_* F)_{s̄} ⟶ (R^q h_* e⁻¹ F)_{x̄} out of the stalk at the image geometric point s̄ = f(x̄). In the situation of Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_iff_forall_isIso_stalk, the base change map f⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F is an isomorphism if and only if that map is an isomorphism for every geometric point x̄ of X.

This is the source's equivalence with each of its two cohomology groups replaced by the stalk which it computes; the identification of the stalks with cohomology over the fibre products Spec(𝒪^{sh}_{S,s̄}) ×_S T and Spec(𝒪^{sh}_{X,x̄}) ×_S T is not available in the library.

public theorem isIso_higherBaseChangeMapAb_app_iff_forall_isIso_stalkMap
    {X Y S T : Scheme.{u}} (f : X ⟶ S) (h : Y ⟶ X) (e : Y ⟶ T) (g : T ⟶ S)
    (sq : IsPullback h e f g) [QuasiCompact g] [QuasiSeparated g]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1}) (q : ℕ) :
    (∀ x : GeometricPoint X,
        IsIso (higherBaseChangeMapAbStalk f g h e sq.w q F x)) ↔
      IsIso ((higherBaseChangeMapAb f g h e sq.w q).app F)

AlgebraicGeometry.isIso_higherBaseChangeMapAb_app_iff_forall_bijective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/StrictHenselizationCriterion.lean, line 630.

Consider a cartesian square of schemes

X  ←-h-  Y
|f       |e
S  ←-g-  T

with g : T ⟶ S quasi-compact and quasi-separated, let F be an abelian sheaf on T_étale and let q ≥ 0. The following are equivalent:

1. for every geometric point x̄ of X with image s̄ = f(x̄), the restriction map H^q(T ×_S Spec(𝒪^{sh}_{S,s̄}), F) ⟶ H^q(T ×_S Spec(𝒪^{sh}_{X,x̄}), F) along the canonical morphism AlgebraicGeometry.strictHenselizationBaseChangeMap is bijective;
2. the base change map f⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F is an isomorphism.

Here the cohomology groups are those of the inverse images of F along the projections to T, and the restriction map is pull back along the canonical morphism together with the comparison of the iterated inverse image with the inverse image along the composite.

theorem isIso_higherBaseChangeMapAb_app_iff_forall_bijective
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1}) (q : ℕ) :
    (∀ x : GeometricPoint X, Function.Bijective
      (etaleCohomologyPullback (strictHenselizationBaseChangeMap B hX hS f g x)
        ((smallEtalePullbackAbComp' _ _ _
          (strictHenselizationBaseChangeMap_fst B hX hS f g x)).hom.app
          F) q)) ↔
      IsIso ((higherBaseChangeMapAb f g h e sq.w q).app F)

Tag 0EZV, a lemma

Tag 0EZV in the Stacks Project.

Stacks.EtaleCohomology.bijective_strictHenselizationFiberRestriction_iff_of_presentation, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationSeparablyClosedFiber.lean, line 560. The attribute's comment: “the equivalence of the tag's two conditions. What is carried as the explicit datum Stacks.EtaleCohomology.SepClosedFiberLocallyConstantPresentation, rather than constructed, is exactly the COHOMOLOGICAL identification of the source's proof: that the two cohomology groups over Spec(O^sh_{S,s̄}) ×_S Spec K are locally constant functions on the limit index space with values in the two groups over Spec K, compatibly with restriction. The geometric input it rests on is proved here, not assumed -- the fibre product is affine (isAffine_strictHenselizationBaseFiber), is Spec of a ring ind-étale over K (indEtale_strictHenselizationBaseFiberRing, from indEtale_etaleLocalRing and Stacks 0BSH), and each étale K-algebra of such a presentation has Spec a finite coproduct of copies of Spec K (specIsoSigmaOfIsSepClosed). Three things block constructing the datum: Theorem 09YQ is available only as the right-derived sheafSections of the small AFFINE étale site and its transport to the Sheaf.H of the small étale site, which is what Stacks.EtaleCohomology.etaleCohomologyRestrictionMap is stated for, needs a general-degree comparison Sheaf.H = rightDerived sheafSections that exists nowhere at these universes; the passage from StacksDraft.IndEtale to an explicit cofiltered limit of schemes is unavailable; and the source's identification of the base change along the morphism of strict henselizations of the different copies of Spec K in a stage -- distinct K-points of Spec(O^sh_{S,s̄}), Galois twists of one another -- is suppressed in the source and not available either. The datum asserts only the cohomological identification, not the equivalence, which is proved here”.

Stacks Project, Tag 0EZV. Let f : X ⟶ S be a morphism of schemes, let x̄ be a geometric point of X with image s̄ in S, let Spec K ⟶ Spec(𝒪^{sh}_{S,s̄}) be a morphism with K a separably closed field, let ℱ be an abelian sheaf on the small étale site of Spec K and let q ≥ 0. Then H^q(Spec(𝒪^{sh}_{X,x̄}) ×_S Spec K, ℱ) = H^q(Spec(𝒪^{sh}_{S,s̄}) ×_S Spec K, ℱ) holds if and only if H^q(Spec(𝒪^{sh}_{X,x̄}) ×_{Spec(𝒪^{sh}_{S,s̄})} Spec K, ℱ) = H^q(Spec K, ℱ) does, each equality read as bijectivity of the canonical restriction map.

The limit presentation of the two cohomology groups over Spec(𝒪^{sh}_{S,s̄}) ×_S Spec K as locally constant functions on a nonempty space is carried as the explicit datum Stacks.EtaleCohomology.SepClosedFiberLocallyConstantPresentation; what is proved here is the equivalence, which comes down to post-composition with a map being bijective on locally constant functions on a nonempty space exactly when the map is.

public theorem bijective_strictHenselizationFiberRestriction_iff_of_presentation
    [IsSepClosed K] (A : Type u) [TopologicalSpace A]
    (F : Sheaf (Spec (CommRingCat.of K)).smallEtaleTopology AddCommGrpCat.{max (u + 1) v})
    (q : ℕ)
    (P : SepClosedFiberLocallyConstantPresentation.{u, v, w} B hX hS f x k A F q) :
    Function.Bijective (strictHenselizationFiberRestriction.{u, v, w} B hX hS f x k F q) ↔
      Function.Bijective (strictHenselizationPointRestriction.{u, v, w} B hX hS f x k F q)

Stacks.EtaleCohomology.bijective_strictHenselizationFiberCohomologyPullback_iff, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationSeparablyClosedFiberCohomology.lean, line 695. The attribute's comment: “corrected: condition (1) is equivalent to condition (2) at every lift k' of Spec K ⟶ S to the strict henselization, not only at the given point; that (2) at the given point does not imply (1) is Stacks.EtaleCohomology.exists_bijective_pointRestriction_and_not_bijective_fiberRestriction”.

[Stacks, Tag 0EZV], corrected. Let f : X ⟶ S be a morphism of schemes, x̄ a geometric point of X with image s̄, k : Spec K ⟶ Spec(𝒪^{sh}_{S,s̄}) with K separably closed, ℱ an abelian sheaf on Spec K and q ≥ 0. Then H^q(Spec(𝒪^{sh}_{X,x̄}) ×_S Spec K, ℱ) = H^q(Spec(𝒪^{sh}_{S,s̄}) ×_S Spec K, ℱ) if and only if H^q(Spec(𝒪^{sh}_{X,x̄}) ×_{Spec(𝒪^{sh}_{S,s̄}), k'} Spec K, ℱ) = H^q(Spec K, ℱ) for every lift k' of the composite Spec K ⟶ S to the strict henselization, each equality read as bijectivity of the restriction map.

public theorem bijective_strictHenselizationFiberCohomologyPullback_iff
    (F : Sheaf (Spec (CommRingCat.of K)).smallEtaleTopology Ab.{u + 1}) (q : ℕ) :
    Function.Bijective (strictHenselizationFiberCohomologyPullback B hX hS f x k F q) ↔
      ∀ k' : Spec (CommRingCat.of K) ⟶ strictHenselization B S hS (x.map f),
        k' ≫ strictHenselizationToBase B S hS (x.map f) =
            k ≫ strictHenselizationToBase B S hS (x.map f) →
          Function.Bijective (strictHenselizationPointCohomologyPullback B hX hS f x k' F q)

Stacks.EtaleCohomology.exists_bijective_pointRestriction_and_not_bijective_fiberRestriction, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationSeparablyClosedFiberCohomology.lean, line 857. The attribute's comment: “the statement as given fails: condition (2) at the given point does not imply condition (1). Condition (1) is equivalent to condition (2) at every lift of Spec K ⟶ S to the strict henselization.”.

Tag 0EZV fails as stated: condition (2) at the given point does not imply condition (1). Let S be the nodal cubic over an algebraically closed field of characteristic zero, s̄ its geometric point at the node, and X = Spec(𝒪^{sh}_{S,s̄}/(e)) for the branch e = y - x √(1 + x) of the node in the strict henselization, with x̄ its geometric point over s̄. For two K-points κ, τ of Spec(𝒪^{sh}_{S,s̄}) over the same point of S, with κ on the branch and τ off it, take the point κ and q = 0, ℱ = ℤ. The fibre of X over κ is Spec K itself, so condition (2) holds; the fibre over τ is empty, so the restriction map of condition (1) kills the class of the constant function 1 on a clopen neighbourhood of τ, and condition (1) fails.

public theorem exists_bijective_pointRestriction_and_not_bijective_fiberRestriction :
    ∃ (B : BigEtaleSite.{u}) (X S : Scheme.{u}) (hX : B.Contains X) (hS : B.Contains S)
      (f : X ⟶ S) (x : GeometricPoint X) (K : Type u) (_ : Field K) (_ : IsAlgClosed K)
      (k : Spec (CommRingCat.of K) ⟶ strictHenselization B S hS (x.map f)),
      Function.Bijective
          (strictHenselizationPointCohomologyPullback B hX hS f x k
            (constantZ (Spec (CommRingCat.of K))) 0) ∧
        ¬ Function.Bijective
          (strictHenselizationFiberCohomologyPullback B hX hS f x k
            (constantZ (Spec (CommRingCat.of K))) 0)

Declarations registered for the tag:

• Stacks.EtaleCohomology.not_forall_bijective_strictHenselizationFiberRestriction_iff, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationFiberErratum.lean, line 68 (StacksFrontier): “erratum: the statement as printed is false; condition (2) at the given point does not imply condition (1). This is the negation of the printed equivalence, for the restriction maps of its statement, at the nodal cubic. The corrected statement, (1) iff (2) for every lift k' of Spec K ⟶ S to the strict henselization, is Stacks.EtaleCohomology.bijective_strictHenselizationFiberCohomologyPullback_iff.”.

Sources

• Section 59.86, Tag 0EZQ
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.etaleLocalBaseChange · Stacks.EtaleCohomology.smallEtaleSetsBaseChange · Stacks.EtaleCohomology.isIso_smallEtaleSetsBaseChange_of_isIso_local · Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_paste · Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange_isIso_of_comp · Stacks.EtaleCohomology.LimitBaseChange.isPullback_limitSquare · Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit · Stacks.EtaleCohomology.LimitBaseChange.isIso_higherBaseChangeMapAb_limit_of_isLimit

Section59.87 Base change for pushforward#

Section 59.87 of the Stacks Project, Tag 0EZW. 6 tags here are carried by 12 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0EZX, a lemma

Tag 0EZX in the Stacks Project.

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_etaleLocalFactorization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChange.lean, line 382. The attribute's comment: “The chain of four identifications of the source's first paragraph -- that the sections of f⁻¹ g_* F and of h_* e⁻¹ F over an object with such a factorization are both the sections of F over V ×_S T, compatibly with the base change map -- rests on Tag 0A3H (used twice, once for f and once for e) read through Tag 03EF, and is carried as the declared hypothesis hlocal. What is supplied here is the source's second paragraph: the covering condition gives enough objects with a flat such factorization, flatness coming from f by hasFlatConnectedFactorization_of_quasiCompact, and a map of sheaves bijective on sections over a family of objects covering every object is an isomorphism.”.

Stacks Project, Tag 0EZX. In a cartesian square of schemes with f : X ⟶ S flat, if every object of the small étale site of X has an étale covering whose members factor through an étale S-scheme by a quasi-compact morphism with geometrically connected fibres, then the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_etaleLocalFactorization (f : X ⟶ S) (h : Y ⟶ X)
    (e : Y ⟶ T) (g : T ⟶ S) (sq : IsPullback h e f g) [Flat f]
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v)))
    (hcover : ∀ U : X.Etale,
      ∃ (ι : Type (u + 1)) (U' : ι → X.Etale) (a : ∀ i, U' i ⟶ U)
        (V : ι → S.Etale) (k : ∀ i, (U' i).left ⟶ (V i).left),
        Sieve.ofArrows U' a ∈ smallEtaleTopology X U ∧
          ∀ i, k i ≫ (V i).hom = (U' i).hom ≫ f ∧
            QuasiCompact (k i) ∧ GeometricallyConnected (k i))
    (hlocal : ∀ ⦃U : X.Etale⦄, HasFlatConnectedFactorization f U →
      Function.Bijective
        (((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F).hom.app (op U))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_etaleLocalFactorizationCover, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeUnconditional.lean, line 850. The attribute's comment: “no residual hypothesis. Both paragraphs of the source's proof are supplied. The first is the chain of four identifications -- that the sections of f⁻¹ g_* F and of h_* e⁻¹ F over an object with such a factorization are both the sections of F over V ×_S T, compatibly with the base change map -- and it is local_baseChange_compatible: Tag 0A3H in the localized form of the source's own argument, used twice, once at f and once at e, is smallEtaleInverseImageSectionsIso and is unconditional, the factorization upstairs is etaleBaseChangeFactor, and their compatibility with the base change map is the Beck-Chevalley identity for the mate defining smallEtaleSetBaseChange. The second paragraph is that the covering condition gives enough objects with a flat such factorization, flatness coming from f by hasFlatConnectedFactorization_of_quasiCompact, and that a map of sheaves bijective on sections over a family of objects covering every object is an isomorphism.”.

In a cartesian square of schemes with f : X ⟶ S flat, if every object of the small étale site of X has an étale covering whose members factor through an étale S-scheme by a quasi-compact morphism with geometrically connected fibres, then the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_etaleLocalFactorizationCover (f : X ⟶ S) (h : Y ⟶ X)
    (e : Y ⟶ T) (g : T ⟶ S) (sq : IsPullback h e f g) [Flat f]
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v)))
    (hcover : ∀ U : X.Etale,
      ∃ (ι : Type (u + 1)) (U' : ι → X.Etale) (a : ∀ i, U' i ⟶ U)
        (V : ι → S.Etale) (k : ∀ i, (U' i).left ⟶ (V i).left),
        Sieve.ofArrows U' a ∈ smallEtaleTopology X U ∧
          ∀ i, k i ≫ (V i).hom = (U' i).hom ≫ f ∧
            QuasiCompact (k i) ∧ GeometricallyConnected (k i)) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Tag 0EYS, a lemma

Tag 0EYS in the Stacks Project.

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_flat_geometricallyReduced, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeUnconditional.lean, line 875. The attribute's comment: “no residual hypothesis. The source merges Tag 0EZX with Tag 0EY5, and both are supplied: the chain of identifications of the sections of f⁻¹ g_* F and of h_* e⁻¹ F over an object carrying a flat quasi-compact factorization with geometrically connected fibres is local_baseChange_compatible, exactly as for Tag 0EZX above, and the covering of the small étale site of X by objects with such a factorization is Tag 0EY5 and is hasEnoughObjects_hasFlatConnectedFactorization. The globalization from sections to sheaves is Tag 0H6Y.”.

In a cartesian square of schemes whose left vertical morphism f : X ⟶ S is flat, locally of finite presentation and has geometrically reduced fibres, the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_flat_geometricallyReduced (f : X ⟶ S) (h : Y ⟶ X)
    (e : Y ⟶ T) (g : T ⟶ S) (sq : IsPullback h e f g)
    [Flat f] [LocallyOfFinitePresentation f] [GeometricallyReduced f]
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Stacks.EtaleCohomology.isIso_smallEtaleBaseChangeAb_of_flat_geometricallyReduced, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmallEtaleBaseChangeAbOfSets.lean, line 303. The attribute's comment: “for abelian sheaves, at the comparison smallEtaleBaseChangeAb of [Stacks, Tag 0F02]; no residual hypothesis. The statement for sheaves of sets is isIso_smallEtaleSetBaseChange_of_flat_geometricallyReduced, and the abelian map is invertible as soon as the set-valued one is, by isIso_smallEtaleBaseChangeAb_of_isIso_smallEtaleSetBaseChange.”.

Stacks Project, Tag 0EYS, for abelian sheaves. In a cartesian square of schemes whose left vertical morphism f : X ⟶ S is flat, locally of finite presentation and has geometrically reduced fibres, the base change map f⁻¹ g_* 𝓕 ⟶ h_* e⁻¹ 𝓕 is an isomorphism for every abelian sheaf 𝓕 on the small étale site of T. The map is Stacks.EtaleCohomology.smallEtaleBaseChangeAb, the underived base change map of [Stacks, Tag 0F02]. It is invertible because the base change map of sheaves of sets is (Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_flat_geometricallyReduced), by Stacks.EtaleCohomology.isIso_smallEtaleBaseChangeAb_of_isIso_smallEtaleSetBaseChange.

public theorem isIso_smallEtaleBaseChangeAb_of_flat_geometricallyReduced (f : X ⟶ S)
    (h : Y ⟶ X) (e : Y ⟶ T) (g : T ⟶ S) (sq : IsPullback h e f g)
    [Flat f] [LocallyOfFinitePresentation f] [GeometricallyReduced f] :
    IsIso (smallEtaleBaseChangeAb f g h e sq.w)

Tag 0EZY, a lemma

Tag 0EZY in the Stacks Project.

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeSepClosedField.lean, line 147. The attribute's comment: “the case of X of finite type over the separably closed field, stated for X locally of finite type. What is proved unconditionally is the covering condition of Tag 0EZX: an affine open of an étale X-scheme is Noetherian, hence locally connected by Tag 04MF, so the connected component of any of its points is an open quasi-compact subscheme, and over a separably closed field a connected scheme is geometrically connected by Tag 0363; flatness of f is automatic over a field. The source's reduction of the general case to this one -- write an affine X as a cofiltered limit of affine finite-type Spec k-schemes and use the compatibility of the base change map with such limits -- is Tag 0EZT part (1), which has no carrier in the library, so the general case is not asserted. The chain of identifications of sections on which Tag 0EZX feeds rests on Tag 0A3H read through Tag 03EF, a separate cited tag unavailable in that form, and is carried -- exactly as in Tags 0EZX, 0EYS and 0EZZ -- as the declared hypothesis hlocal.”.

Stacks Project, Tag 0EZY, in the case of X locally of finite type over the separably closed field. Let k be a separably closed field and let f : X ⟶ Spec k be locally of finite type. Then in a cartesian square over f the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_isSepClosed (k : Type u) [Field k] [IsSepClosed k]
    (f : X ⟶ Spec (.of k)) [LocallyOfFiniteType f] (h : Y ⟶ X) (e : Y ⟶ T)
    (g : T ⟶ Spec (.of k)) (sq : IsPullback h e f g)
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v)))
    (hlocal : ∀ ⦃U : X.Etale⦄, HasFlatConnectedFactorization f U →
      Function.Bijective
        (((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F).hom.app (op U))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed_of_quasiCompact, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeSepClosedFieldGeneral.lean, line 549. The attribute's comment: “the tag's statement for an arbitrary scheme X over the spectrum of a separably closed field, with nothing assumed of X and no result of the source assumed. The assumption that g be quasi-compact and quasi-separated is the standing convention of Section 0EZW, stated at its start for every cartesian diagram of the section, and the source's proof passes through Tag 0EZT, which assumes it; it cannot be dropped (see this file's module docstring for a counterexample with g not quasi-compact). The proof is the stalkwise one of Tag 0F00 specialized to S = Spec k: stalks at geometric points detect isomorphisms; the stalk at x̄ is global sections over the strict henselization Z = Spec(O^sh_{X,x̄}); the pasting law for the base change maps of the square of Z ⟶ X stacked on the given square splits the base change map of the outer rectangle into Γ(Z, ι⁻¹ β) followed by the base change map of the square of Z ⟶ X over h; the latter is bijective on global sections by Tag 03Q9 for sheaves of sets (isIso_Γ_map_smallEtaleSetBaseChange_strictHenselization, h being quasi-compact and quasi-separated as a base change of g), and the former composite is bijective by the first paragraph of Tag 0EZX (local_baseChange_compatible) at the final object of Z_étale, Z ⟶ Spec k being flat, quasi-compact and geometrically connected (Z is local, hence connected, and connected implies geometrically connected over a separably closed field, Tag 0363). The source's reduction to X affine of finite type through Tags 0EZR and 0EZT is replaced by this: its limit argument is the one inside Tag 03Q9.”.

Let k be a separably closed field and consider a cartesian square of schemes

X <--h-- Y
|f       |e
Spec k <--g-- T

with g quasi-compact and quasi-separated. Then the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T. Nothing is assumed of X.

theorem isIso_smallEtaleSetBaseChange_of_isSepClosed_of_quasiCompact
    (sq : IsPullback h e f g) [QuasiCompact g] [QuasiSeparated g]
    (F : Sheaf (smallEtaleTopology T) (Type (u + 1))) :
    IsIso ((smallEtaleSetBaseChange.{u, 0} e h g f sq.w.symm).app F)

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed_locallyOfFiniteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeSepClosedFieldUnconditional.lean, line 96. The attribute's comment: “the case of X of finite type over the separably closed field, stated for X locally of finite type. What is proved unconditionally is the covering condition of Tag 0EZX: an affine open of an étale X-scheme is Noetherian, hence locally connected by Tag 04MF, so the connected component of any of its points is an open quasi-compact subscheme, and over a separably closed field a connected scheme is geometrically connected by Tag 0363; flatness of f is automatic over a field. The chain of identifications of sections on which Tag 0EZX feeds is supplied too, by local_baseChange_compatible, exactly as in Tags 0EZX, 0EYS and 0EZZ. What is not asserted is the source's reduction of the general case to this one, which takes two steps. Working locally on X to reduce to an affine X is Tag 0EZR clause (1), and that clause is now proved, as isIso_smallEtaleSetsBaseChange_of_isIso_local; but it is proved about smallEtaleSetsBaseChange, a second set-valued base change map built in BaseChangeHigher/OrdinaryLocality.lean from the composition isomorphisms of direct images, which no result identifies with the smallEtaleSetBaseChange of this square. The second step writes an affine X as a cofiltered limit of affine finite-type Spec k-schemes and uses the compatibility of the base change map with such limits, which is Tag 0EZT part (1); the two 0EZT carriers give the cartesian square on the limits and part (2), for abelian coefficients and the higher direct images, and part (1) -- sheaves of sets and the underived direct image, the form this map takes -- has no carrier, nor has the set-valued colimit assertion of Tag 0EYM it would be built from. The hypothesis that f be locally of finite type is therefore still needed, and is more than the source's statement assumes.”.

Stacks Project, Tag 0EZY, in the case of X locally of finite type over the separably closed field. Let k be a separably closed field and let f : X ⟶ Spec k be locally of finite type. Then in a cartesian square over f the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_isSepClosed_locallyOfFiniteType (k : Type u) [Field k]
    [IsSepClosed k] (f : X ⟶ Spec (.of k)) [LocallyOfFiniteType f] (h : Y ⟶ X) (e : Y ⟶ T)
    (g : T ⟶ Spec (.of k)) (sq : IsPullback h e f g)
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Tag 0EZZ, a lemma

Tag 0EZZ in the Stacks Project.

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isOpenMap_connectedFibreOpens, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeUnconditional.lean, line 902. The attribute's comment: “no residual hypothesis. The source's proof is omitted with the hint that the hypotheses imply the covering condition of Tag 0EZX; that implication is what is proved here, and the chain of identifications of sections which Tag 0EZX then feeds on is local_baseChange_compatible, exactly as for Tag 0EZX above. Hypothesis (3) of the source is read as the existence of an isomorphism of the étale X-scheme with a finite coproduct of open subschemes of X over X; the parenthetical examples in (3) and (4), which the source derives from Tag 09Z9, are not asserted.”.

Let f : X ⟶ S be flat and open, with separably closed residue fields on S, with every affine étale X-scheme a finite disjoint union of open subschemes of X, and with every nonempty open of a fibre of f connected. Then in a cartesian square over f the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on the small étale site of T.

theorem isIso_smallEtaleSetBaseChange_of_isOpenMap_connectedFibreOpens (f : X ⟶ S) (h : Y ⟶ X)
    (e : Y ⟶ T) (g : T ⟶ S) (sq : IsPullback h e f g) [Flat f] (hopen : IsOpenMap f)
    (hsep : ∀ s : S, IsSepClosed (S.residueField s))
    (hsplit : ∀ U : X.Etale, IsAffine U.left →
      ∃ (n : ℕ) (V : Fin n → Scheme.{u}) (j : ∀ a, V a ⟶ X)
        (_ : ∀ a, IsOpenImmersion (j a)) (i : U.left ≅ ∐ V),
        i.hom ≫ Sigma.desc j = U.hom)
    (hconn : ∀ s : S, ∀ W : (f.fiber s).Opens, W ≠ ⊥ → ConnectedSpace W.toScheme)
    (F : Sheaf (smallEtaleTopology T) (Type (max (u + 1) v))) :
    IsIso ((smallEtaleSetBaseChange.{u, v} e h g f sq.w.symm).app F)

Tag 0EYR, a lemma

Tag 0EYR in the Stacks Project.

Stacks.EtaleCohomology.preservesLimitsOfShape_smallEtaleInverseImage_of_geometricallyReduced, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/PushforwardBaseChangeUnconditional.lean, line 107. The attribute's comment: “no residual hypothesis. Both inputs of the source's proof are supplied: the covering of the small étale site by objects with a flat quasi-compact factorization with geometrically connected fibres, which is Tag 0EY5 and is hasEnoughObjects_hasFlatConnectedFactorization; and the bijectivity of the canonical comparison G(V) ⟶ (f⁻¹G)(U) over such an object, which is Tag 0A3H in the localized form of the source's own argument and is smallEtaleSectionsComparisonIsIso, proved from Tag 0EZK and the small-étale instance of Tag 03EF. The globalization from sections to sheaves is Tag 0H6Y.”.

For a morphism of schemes which is flat, locally of finite presentation and has geometrically reduced fibres, the inverse image functor on sheaves of sets of the small étale sites commutes with products.

theorem preservesLimitsOfShape_smallEtaleInverseImage_of_geometricallyReduced
    [Flat f] [LocallyOfFinitePresentation f] [GeometricallyReduced f]
    (I : Type w) :
    PreservesLimitsOfShape (Discrete I) (smallEtaleInverseImage.{u, max v w} f)

Tag 0F00, a lemma

Tag 0F00 in the Stacks Project.

Stacks.EtaleCohomology.isPullback_strictHenselizationPullbackMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationBaseChangeSections.lean, line 131. The attribute's comment: “the identity Spec(O^sh_{X,x̄}) ×_X Y = Spec(O^sh_{S,f(x̄)}) ×_S T-base-change of Spec(O^sh_{X,x̄}) ⟶ Spec(O^sh_{S,f(x̄)}) of the source's last sentence, and nothing else. The tag's conclusion f⁻¹ g_* F = h_* e⁻¹ F is NOT asserted here or in globalSectionsIso_strictHenselizationPullback below: the source obtains it from this identification together with Tag 03PU and Tag 03Q9, and Tag 03Q9 for sheaves of sets is not available in the library (the only form of it present is restricted to a base which is already the spectrum of a strictly henselian local ring).”.

Stacks Project, Tag 0F00, the base change step of its proof. In a cartesian square with h : Y ⟶ X, e : Y ⟶ T and g : T ⟶ S, the canonical morphism Spec(O^sh_{X,x̄}) ×_X Y ⟶ Spec(O^sh_{S,f(x̄)}) ×_S T is a base change of the morphism of strict henselizations Spec(O^sh_{X,x̄}) ⟶ Spec(O^sh_{S,f(x̄)}).

theorem isPullback_strictHenselizationPullbackMap (sq : IsPullback h e f g) :
    IsPullback (pullback.fst (strictHenselizationToBase B X hX x) h)
      (strictHenselizationPullbackMap B hX hS f x sq)
      (strictHenselizationSquareMap B hX hS f x)
      (pullback.fst (strictHenselizationToBase B S hS (x.map f)) g)

Stacks.EtaleCohomology.globalSectionsIso_strictHenselizationPullback, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationBaseChangeSections.lean, line 162. The attribute's comment: “the application of Tag 0A3H to the morphism Spec(O^sh_{X,x̄}) ×_X Y ⟶ Spec(O^sh_{S,f(x̄)}) ×_S T with which the source's proof concludes, and nothing else. The tag's conclusion f⁻¹ g_* F = h_* e⁻¹ F is NOT asserted: the source reads the two sides of this identification as the stalks at x̄ of f⁻¹ g_* F and of h_* e⁻¹ F by Tag 03Q9 and concludes by Tag 03PU, and Tag 03Q9 for sheaves of sets is absent from the library. Flatness of the morphism of strict henselizations, which Tag 0A3H needs and the source leaves implicit in the flatness of f, is the declared hypothesis hflat.”.

Stacks Project, Tag 0F00, the application of Tag 0A3H in its proof. If the morphism of strict henselizations at a geometric point x̄ of X is flat with geometrically connected fibres, then in a cartesian square the global sections of a sheaf of sets on the small étale site of Spec(O^sh_{S,f(x̄)}) ×_S T are the global sections of its inverse image on the small étale site of Spec(O^sh_{X,x̄}) ×_X Y, naturally in the sheaf.

def globalSectionsIso_strictHenselizationPullback (sq : IsPullback h e f g)
    (hflat : Flat (strictHenselizationSquareMap B hX hS f x))
    (hconn : GeometricallyConnected (strictHenselizationSquareMap B hX hS f x)) :
    Sheaf.Γ (smallEtaleTopology (strictHenselizationPullback B hS (x.map f) g))
        (Type (max (u + 1) v)) ≅
      smallEtaleInverseImage.{u, v} (strictHenselizationPullbackMap B hX hS f x sq) ⋙
        Sheaf.Γ (smallEtaleTopology (strictHenselizationPullback B hX x h))
          (Type (max (u + 1) v))

Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_geometricallyConnected_strictHenselization, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/StrictHenselizationBaseChangeUnconditional.lean, line 296.

Let f : X ⟶ S be flat, and suppose that for every geometric point x̄ of X the morphism Spec(O^sh_{X,x̄}) ⟶ Spec(O^sh_{S,f(x̄)}) of strict henselizations has geometrically connected fibres. Then for every cartesian square

Y --h--> X
|e       |f
v        v
T --g--> S

with g quasi-compact and quasi-separated, the canonical base change map f⁻¹ g_* F ⟶ h_* e⁻¹ F is an isomorphism for every sheaf of sets F on T_étale.

The strict henselizations are those constructed inside a big étale site B containing X and S; the base change map is Stacks.EtaleCohomology.smallEtaleSetBaseChange, and the conclusion is that it is an isomorphism of functors, that is, at every F.

theorem isIso_smallEtaleSetBaseChange_of_geometricallyConnected_strictHenselization [Flat f]
    (hconn : ∀ x : GeometricPoint X,
      GeometricallyConnected (strictHenselizationSquareMap B hX hS f x))
    (sq : IsPullback h e f g) [QuasiCompact g] [QuasiSeparated g] :
    IsIso (smallEtaleSetBaseChange.{u, 0} e h g f sq.w.symm)

Sources

• Section 59.87, Tag 0EZW
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_etaleLocalFactorization · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_etaleLocalFactorizationCover · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_flat_geometricallyReduced · Stacks.EtaleCohomology.isIso_smallEtaleBaseChangeAb_of_flat_geometricallyReduced · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed_of_quasiCompact · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isSepClosed_locallyOfFiniteType · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isOpenMap_connectedFibreOpens

Section59.88 Base change for higher direct images#

Section 59.88 of the Stacks Project, Tag 0F01. 7 tags here are carried by 14 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F02, a remark

Tag 0F02 in the Stacks Project.

Stacks.EtaleCohomology.higherBaseChangeMapAb, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/BaseChangeCondition.lean, line 277. The attribute's comment: “bc”.

The base change map on higher direct images, (f')⁻¹ R^q g_* ⟶ R^q h_* e⁻¹, for abelian sheaves on the small étale sites of a commutative square. It is the degree-q cohomology of the canonical derived base change comparison smallEtaleDerivedBaseChangeAb, transported along the canonical identifications baseChangeSourceIso and baseChangeTargetIso of the two sides with cohomology of the bounded below derived composites.

public def higherBaseChangeMapAb (q : ℕ) :
    (smallEtalePushforwardAb g).rightDerived q ⋙ smallEtalePullbackAb f' ⟶
      smallEtalePullbackAb e ⋙ (smallEtalePushforwardAb h).rightDerived q

Stacks.EtaleCohomology.BaseChangeHolds, an abbrev in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/BaseChangeCondition.lean, line 352.

The condition BC(f, n, q₀) of Stacks 0F02. For f : X ⟶ S a morphism of schemes and n an integer, BC(f, n, q₀) holds if for every commutative diagram

X  ←--  X'  ←-h-  Y
|f      |f'       |e
S  ←--  S'  ←-g-  T

in which X' = X ×_S S' and Y = X' ×_{S'} T and g is quasi-compact and quasi-separated, and every abelian sheaf F on the small étale site of T annihilated by n, the base change map (f')⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F is an isomorphism for every q ≤ q₀.

The two cartesian squares are the hypotheses IsPullback p f' f s and IsPullback h e f' g; the map whose invertibility is asserted is higherBaseChangeMapAb, the base change map on higher direct images. Annihilation by n is imposed on every section of F, which is the same as annihilation of F itself.

public abbrev BaseChangeHolds {X S : Scheme.{u}} (f : X ⟶ S) (n q₀ : ℤ) : Prop

Tag 0F03, a lemma

Tag 0F03 in the Stacks Project.

Stacks.EtaleCohomology.mono_cokernelDesc_higherBaseChangeMapAb_app_of_injective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/TorsionSheafEmbedding.lean, line 281. The attribute's comment: “part (3), the cokernel inside R^q h_* e⁻¹ of an injective ℤ/nℤ-module sheaf”.

Part (3) of [Stacks, Tag 0F03]. In the situation of mono_higherBaseChangeMapAb_app_of_injective, with F annihilated by n contained in the underlying abelian sheaf of an injective sheaf I of ℤ/nℤ-modules,

Coker((f')⁻¹R^{q+1} g_* F ⟶ R^{q+1} h_* e⁻¹F) ⊂ R^{q+1} h_* e⁻¹I,

the inclusion being the canonical map induced by R^{q+1}h_* e⁻¹(F ⟶ I), which is defined because higherBaseChangeMapAb_app_comp_map_eq_zero_of_injective vanishes. This is mono_cokernelDesc_higherBaseChangeMapAb_app_of_shortExact with its short exact sequence constructed from F ⊂ I.

public theorem mono_cokernelDesc_higherBaseChangeMapAb_app_of_injective
    (hbc : BaseChangeHolds f n (q : ℤ))
    (sq₁ : IsPullback p f' f s) (sq : IsPullback h e f' g)
    [QuasiCompact g] [QuasiSeparated g]
    {F : Sheaf (smallEtaleTopology T) Ab.{u + 1}}
    (hF : ∀ (U : T.Etale) (x : F.obj.obj (op U)), n • x = 0)
    (I : SheafOfModules.{u + 1} (etaleZModRingSheaf T n)) [Injective I]
    (k : F ⟶ (SheafOfModules.toSheaf.{u + 1} (etaleZModRingSheaf T n)).obj I) (hk : Mono k) :
    Mono (cokernel.desc ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).app F)
      ((higherBaseChangeTarget h e (q + 1)).map k)
      (higherBaseChangeMapAb_app_comp_map_eq_zero_of_injective n q f' g h e sq I k hk))

Stacks.EtaleCohomology.mono_higherBaseChangeMapAb_app_of_annihilated, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/TorsionSheafEmbedding.lean, line 355. The attribute's comment: “part (1), injectivity of the base change map”.

Part (1) of [Stacks, Tag 0F03]. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q), let

X  ←--  X'  ←-h-  Y
|f      |f'       |e
S  ←--  S'  ←-g-  T

be a commutative diagram with X' = X ×_S S' and Y = X' ×_{S'} T and g quasi-compact and quasi-separated, and let F be an abelian sheaf on the small étale site of T annihilated by n. Then the base change map (f')⁻¹R^{q+1} g_* F ⟶ R^{q+1} h_* e⁻¹F is injective.

This is mono_higherBaseChangeMapAb_app_of_injective at the embedding torsionEtaleInjectiveEmbedding of F into an injective sheaf of ℤ/nℤ-modules.

public theorem mono_higherBaseChangeMapAb_app_of_annihilated
    (hbc : BaseChangeHolds f n (q : ℤ))
    (sq₁ : IsPullback p f' f s) (sq : IsPullback h e f' g)
    [QuasiCompact g] [QuasiSeparated g]
    {F : Sheaf (smallEtaleTopology T) Ab.{u + 1}}
    (hF : ∀ (U : T.Etale) (x : F.obj.obj (op U)), n • x = 0) :
    Mono ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).app F)

Stacks.EtaleCohomology.mono_cokernelMap_higherBaseChangeMapAb_app_of_annihilated, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/TorsionSheafEmbedding.lean, line 377. The attribute's comment: “part (2), the cokernel inclusion under F ⊂ G”.

Part (2) of [Stacks, Tag 0F03]. In the situation of mono_higherBaseChangeMapAb_app_of_annihilated, if F ⊂ G where G on the small étale site of T is annihilated by n, then

Coker((f')⁻¹R^{q+1}g_* F → R^{q+1}h_* e⁻¹F) ⊂ Coker((f')⁻¹R^{q+1}g_* G → R^{q+1}h_* e⁻¹G),

the inclusion being the map induced on cokernels by the naturality square of the base change map. This is mono_cokernelMap_higherBaseChangeMapAb_app_of_injective at the embedding torsionEtaleInjectiveEmbedding of G into an injective sheaf of ℤ/nℤ-modules, which realises the choice F ⊂ G ⊂ I of the source.

public theorem mono_cokernelMap_higherBaseChangeMapAb_app_of_annihilated
    (hbc : BaseChangeHolds f n (q : ℤ))
    (sq₁ : IsPullback p f' f s) (sq : IsPullback h e f' g)
    [QuasiCompact g] [QuasiSeparated g]
    {F G : Sheaf (smallEtaleTopology T) Ab.{u + 1}} (i : F ⟶ G) (hi : Mono i)
    (hF : ∀ (U : T.Etale) (x : F.obj.obj (op U)), n • x = 0)
    (hG : ∀ (U : T.Etale) (x : G.obj.obj (op U)), n • x = 0) :
    Mono (cokernel.map ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).app F)
      ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).app G)
      ((higherBaseChangeSource f' g (q + 1)).map i)
      ((higherBaseChangeTarget h e (q + 1)).map i)
      ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).naturality i).symm)

Tag 0F04, a lemma

Tag 0F04 in the Stacks Project.

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_isIso_integralSurjective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/IntegralDirectImagePasting.lean, line 235.

Stacks Project, Tag 0F04. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q). Consider the commutative diagrams

X  ←--  X'  ←-h-  Y  ←-π'-  Y'            X'  ←----h' = π' ≫ h----  Y'
|f      |f'       |e        |e'    and    |f'                       |e'
S  ←--  S'  ←-g-  T  ←-π-   T'            S'  ←----g' = π ≫ g-----  T'

with all squares cartesian, g quasi-compact and quasi-separated and π integral surjective, and let F be an abelian sheaf on the small étale site of T annihilated by n, with F' = π⁻¹F. If the base change map (f')⁻¹ R^{q+1} g'_* F' ⟶ R^{q+1} h'_* (e')⁻¹ F' of the pasted diagram is an isomorphism, then so is the base change map (f')⁻¹ R^{q+1} g_* F ⟶ R^{q+1} h_* e⁻¹ F of the middle square.

The index convention is that of [Stacks, Tag 0F03] in this library: the source's q ≥ 1 with BC(f, n, q - 1) is read as degree q + 1 with BC(f, n, q).

The unit F ⟶ π_*π⁻¹F is a monomorphism because π is surjective (Stacks.EtaleCohomology.mono_smallEtaleUnitAb_of_surjective) and π_*π⁻¹F is annihilated by n (Stacks.EtaleCohomology.zsmul_eq_zero_of_additive), so by parts (1) and (2) of [Stacks, Tag 0F03] it suffices that the base change map of the middle square at π_*π⁻¹F is an isomorphism. That is the hypothesis, read through Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_pushforward_iff, which is where [Stacks, Tag 0EYP], [Stacks, Tag 04C2] and the relative Leray identification Rπ_* = π_* enter.

public theorem isIso_higherBaseChangeMapAb_app_of_isIso_integralSurjective
    {X S S' X' T Y T' Y' : Scheme.{u}} (f : X ⟶ S) (n : ℤ) (q : ℕ)
    (hbc : BaseChangeHolds f n (q : ℤ))
    (s : S' ⟶ S) (p : X' ⟶ X) (f' : X' ⟶ S') (sq₁ : IsPullback p f' f s)
    (g : T ⟶ S') (h : Y ⟶ X') (e : Y ⟶ T) (sq₂ : IsPullback h e f' g)
    [QuasiCompact g] [QuasiSeparated g]
    (π : T' ⟶ T) (π' : Y' ⟶ Y) (e' : Y' ⟶ T') (sq₃ : IsPullback π' e' e π)
    [IsIntegralHom π] [Surjective π]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (hF : ∀ (U : T.Etale) (y : F.obj.obj (op U)), n • y = 0)
    (hpaste : IsIso ((higherBaseChangeMapAb f' (π ≫ g) (π' ≫ h) e'
      (sq₃.paste_horiz sq₂).w (q + 1)).app ((smallEtalePullbackAb π).obj F))) :
    IsIso ((higherBaseChangeMapAb f' g h e sq₂.w (q + 1)).app F)

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_integralSurjective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/IntegralSurjectiveCoverCriterion.lean, line 145.

Stacks Project, Tag 0F04. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q). Consider the commutative diagrams

X  ←--  X'  ←-h-  Y  ←-π'-  Y'            X'  ←----h' = π' ≫ h----  Y'
|f      |f'       |e        |e'    and    |f'                       |e'
S  ←--  S'  ←-g-  T  ←-π-   T'            S'  ←----g' = π ≫ g-----  T'

with all squares cartesian, g quasi-compact and quasi-separated and π integral surjective, and let F be an abelian sheaf on the small étale site of T annihilated by n, with F' = π⁻¹F. If the base change map (f')⁻¹ R^{q+1} g'_* F' ⟶ R^{q+1} h'_* (e')⁻¹ F' of the pasted diagram is an isomorphism, then so is the base change map (f')⁻¹ R^{q+1} g_* F ⟶ R^{q+1} h_* e⁻¹ F of the middle square.

The index convention is that of [Stacks, Tag 0F03] in this library: the source's q ≥ 1 with BC(f, n, q - 1) is read as degree q + 1 with BC(f, n, q).

Both base change maps are decided by the criterion isIso_higherBaseChangeMapAb_app_iff_map_eq_zero of [Stacks, Tag 0F03]. Embed F' into an injective sheaf J' of ℤ/nℤ-modules by k'; the hypothesis says R^{q+1}h'_*(e')⁻¹k' = 0. The sheaf π_*J' is annihilated by n and receives F by the monomorphism F ⟶ π_*π⁻¹F ⟶ π_*J', the unit being injective because π is surjective; embedding π_*J' into an injective sheaf of ℤ/nℤ-modules gives an embedding of F factoring through it. And R^{q+1}h_* e⁻¹(π_*k') vanishes: through the base change transformation e⁻¹π_* ⟶ π'_*(e')⁻¹ of exists_baseChange_isIso_of_baseChangeHolds, invertible at J', it is carried to R^{q+1}h_*(π'_*(e')⁻¹k'), which is R^{q+1}h'_*(e')⁻¹k' read through integralCompRightDerivedIso π' h, π' being integral.

public theorem isIso_higherBaseChangeMapAb_app_of_integralSurjective
    {X S S' X' T Y T' Y' : Scheme.{u}} (f : X ⟶ S) (n : ℤ) (q : ℕ)
    (hbc : BaseChangeHolds f n (q : ℤ))
    (s : S' ⟶ S) (p : X' ⟶ X) (f' : X' ⟶ S') (sq₁ : IsPullback p f' f s)
    (g : T ⟶ S') (h : Y ⟶ X') (e : Y ⟶ T) (sq₂ : IsPullback h e f' g)
    [QuasiCompact g] [QuasiSeparated g]
    (π : T' ⟶ T) (π' : Y' ⟶ Y) (e' : Y' ⟶ T') (sq₃ : IsPullback π' e' e π)
    [IsIntegralHom π] [Surjective π]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (hF : ∀ (U : T.Etale) (y : F.obj.obj (op U)), n • y = 0)
    (hcollapse : IsIso ((higherBaseChangeMapAb f' (π ≫ g) (π' ≫ h) e'
      (sq₃.paste_horiz sq₂).w (q + 1)).app ((smallEtalePullbackAb π).obj F))) :
    IsIso ((higherBaseChangeMapAb f' g h e sq₂.w (q + 1)).app F)

Tag 0F05, a lemma

Tag 0F05 in the Stacks Project.

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_iff_of_isIntegralHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/IntegralBaseCriterion.lean, line 98. The attribute's comment: “as an equivalence; the source's statement is the forward implication”.

Stacks Project, Tag 0F05. With f : X ⟶ S and n as in [Stacks, Tag 0F02], let q ≥ 1 and assume BC(f, n, q - 1). Consider the commutative diagrams

X  ←--  X'  ←-π'-  X''  ←-h'-  Y                X'  ←------h------  Y
|f      |f'        |f''        |e       and     |f'                 |e
S  ←--  S'  ←-π-   S''  ←-g'-  T                S'  ←------g------  T

with all squares cartesian, g' quasi-compact and quasi-separated, π integral, h = h' ≫ π' and g = g' ≫ π, and let F be an abelian sheaf on the small étale site of T annihilated by n. Then the base change map (f')⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F of the collapsed diagram is an isomorphism if and only if the base change map (f'')⁻¹ R^q g'_* F ⟶ R^q h'_* e⁻¹ F of the right-hand square is. The source's statement is the forward implication.

Both are decided by the criterion isIso_higherBaseChangeMapAb_app_iff_map_eq_zero at one embedding F ⊂ I into an injective sheaf of ℤ/nℤ-modules, and the two morphisms it asks to vanish correspond under rightDerivedCompIntegralIso h' π', the faithful functor π'_* carrying one to the other.

public theorem isIso_higherBaseChangeMapAb_app_iff_of_isIntegralHom
    {X S X' S' X'' S'' Y T : Scheme.{u}} (f : X ⟶ S) (n : ℤ) (q : ℕ) (hq : 1 ≤ q)
    (hbc : BaseChangeHolds f n ((q : ℤ) - 1)) (s : S' ⟶ S) (p : X' ⟶ X) (f' : X' ⟶ S')
    (sq₁ : IsPullback p f' f s) (π : S'' ⟶ S') (π' : X'' ⟶ X') (f'' : X'' ⟶ S'')
    (sq₂ : IsPullback π' f'' f' π) (g' : T ⟶ S'') (h' : Y ⟶ X'') (e : Y ⟶ T)
    (sq₃ : IsPullback h' e f'' g') [QuasiCompact g'] [QuasiSeparated g'] [IsIntegralHom π]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (hF : ∀ (U : T.Etale) (x : F.obj.obj (Opposite.op U)), n • x = 0) :
    IsIso ((higherBaseChangeMapAb f' (g' ≫ π) (h' ≫ π') e (sq₃.paste_horiz sq₂).w q).app F) ↔
      IsIso ((higherBaseChangeMapAb f'' g' h' e sq₃.w q).app F)

Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_isIso_collapsed_of_bc, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/IntegralBaseLeray.lean, line 192.

Stacks Project, Tag 0F05. With f : X ⟶ S and n as in [Stacks, Tag 0F02], let q ≥ 1 and assume BC(f, n, q - 1). Consider the commutative diagrams

X  ←--  X'  ←-π'-  X''  ←-h'-  Y                X'  ←------h------  Y
|f      |f'        |f''        |e       and     |f'                 |e
S  ←--  S'  ←-π-   S''  ←-g'-  T                S'  ←------g------  T

with all squares cartesian, g' quasi-compact and quasi-separated, π integral, h = h' ≫ π' and g = g' ≫ π, and let F be an abelian sheaf on the small étale site of T annihilated by n. If the base change map (f')⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F of the collapsed diagram is an isomorphism, then so is the base change map (f'')⁻¹ R^q g'_* F ⟶ R^q h'_* e⁻¹ F of the right-hand square.

The two relative Leray identifications the source's proof uses are supplied, not assumed: they are Stacks.EtaleCohomology.integralLerayIso for g = g' ≫ π and for h = h' ≫ π', the second along the base change π' of π, whose integrality is Stacks.EtaleCohomology.isIntegralHom_of_isPullback.

The remaining identification, [Stacks, Tag 0EYP] for the middle square, is not assumed either: it is needed only at the cohomology sheaves of Rg'_* F, which are annihilated by n because F is, and there it is the source's own BC(f, n, q - 1) read in degree zero, which is Stacks.EtaleCohomology.isIso_smallEtaleBaseChangeAb_app_of_annihilated. That only the cohomology sheaves matter is Stacks.Cohomology.BaseChange.isIso_derived_app_iff_homology, both legs of the middle square's direct image being exact.

The pasting compatibility the source elides is not assumed: it is Stacks.EtaleCohomology.smallEtaleDerivedBaseChangeAb_eq_derivedVPaste_app, the pasting law for the derived base change comparison along the direct image legs, whose two leg comparisons Rg'_* ⋙ Rπ_* ≅ R(g' ≫ π)_* and Rh'_* ⋙ Rπ'_* ≅ R(h' ≫ π')_* are the derived form of the relative Leray identification for an integral morphism. Of the four factors of that pasting, two are isomorphisms outright, the third is one by the preceding paragraph, and the fourth is the direct image along π' of the base change map of the right-hand square; the conclusion is then isIso_pushforward_higherBaseChangeMapAb_app_iff in the form carried by Stacks.EtaleCohomology.isIso_smallEtalePushforwardAb_map_iff_of_isIntegralHom.

public theorem isIso_higherBaseChangeMapAb_app_of_isIso_collapsed_of_bc
    {X S X' S' X'' S'' Y T : Scheme.{u}} (f : X ⟶ S) (n : ℤ) (q : ℕ) (hq : 1 ≤ q)
    (hbc : BaseChangeHolds f n ((q : ℤ) - 1)) (s : S' ⟶ S) (p : X' ⟶ X) (f' : X' ⟶ S')
    (sq₁ : IsPullback p f' f s) (π : S'' ⟶ S') (π' : X'' ⟶ X') (f'' : X'' ⟶ S'')
    (sq₂ : IsPullback π' f'' f' π) (g' : T ⟶ S'') (h' : Y ⟶ X'') (e : Y ⟶ T)
    (sq₃ : IsPullback h' e f'' g') [QuasiCompact g'] [QuasiSeparated g'] [IsIntegralHom π]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (hF : ∀ (U : T.Etale) (x : F.obj.obj (Opposite.op U)), n • x = 0)
    (hcollapse : IsIso
      ((higherBaseChangeMapAb f' (g' ≫ π) (h' ≫ π') e (sq₃.paste_horiz sq₂).w q).app F)) :
    IsIso ((higherBaseChangeMapAb f'' g' h' e sq₃.w q).app F)

Tag 0F06, a lemma

Tag 0F06 in the Stacks Project.

AlgebraicGeometry.exists_field_point, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/FieldPoint.lean, line 559.

Stacks Project, Tag 0F06. A property of quasi-compact, quasi-separated schemes over a nonempty base, invariant under thickenings and stable under affine-transition inverse limits and passage to a closed subscheme or its quasi-compact open complement, holds at some field-valued point.

The [Nonempty T] hypothesis is deliberate and necessary rather than a weakening. Without it the statement is false: T = Spec PUnit is empty and affine, hence quasi-compact and quasi-separated, and for P = fun _ _ ↦ True all four remaining hypotheses hold, while the conclusion would supply a morphism into T from the nonempty scheme Spec K. The Stacks statement leaves the condition implicit; the proof below needs it to put ⊥ into the family of closed subschemes that are nonempty and carry P, which is where Zorn's lemma is applied.

public theorem exists_field_point {T : Scheme.{u}} [Nonempty T] [CompactSpace T]
    [QuasiSeparatedSpace T] (P : ∀ X : Scheme.{u}, (X ⟶ T) → Prop)
    (hThickening : ∀ {X X' : Scheme.{u}} [CompactSpace X] [QuasiSeparatedSpace X]
      [CompactSpace X'] [QuasiSeparatedSpace X'] (i : X ⟶ X') (f : X' ⟶ T),
      IsThickening i → (P X (i ≫ f) ↔ P X' f))
    (hLimit : ∀ {I : Type u} [Category.{u} I] [IsCofiltered I]
      (D : I ⥤ Scheme.{u})
      [∀ {i j : I} (a : i ⟶ j), IsAffineHom (D.map a)]
      [∀ i, CompactSpace (D.obj i)] [∀ i, QuasiSeparatedSpace (D.obj i)]
      (c : Cone D) (_ : IsLimit c)
      [CompactSpace c.pt] [QuasiSeparatedSpace c.pt]
      (f : c.pt ⟶ T) (fi : ∀ i, D.obj i ⟶ T),
      (∀ {i j : I} (a : i ⟶ j), D.map a ≫ fi j = fi i) →
      (∀ i, c.π.app i ≫ fi i = f) →
      (∀ i, P (D.obj i) (fi i)) → P c.pt f)
    (hClosedOrOpen : ∀ {X : Scheme.{u}} [CompactSpace X]
      [QuasiSeparatedSpace X] (f : X ⟶ T) (J : X.IdealSheafData)
      [CompactSpace J.subscheme] [QuasiSeparatedSpace J.subscheme]
      (V : X.Opens) (_hV : (V : Set X) = (J.support : Set X)ᶜ)
      [CompactSpace V.toScheme] [QuasiSeparatedSpace V.toScheme],
      P X f → (P V.toScheme (V.ι ≫ f) ∨
        P J.subscheme (J.subschemeι ≫ f)))
    (hP : P T (𝟙 T)) :
    ∃ (K : Type u) (_ : Field K) (g : Spec (.of K) ⟶ T),
      P (Spec (.of K)) g

AlgebraicGeometry.exists_field_point_of_isDirected, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/FieldPointDirected.lean, line 108. The attribute's comment: “the limit hypothesis for directed inverse systems, as in the source”.

Stacks Project, Tag 0F06. Let T be a nonempty quasi-compact quasi-separated scheme and P a property of quasi-compact quasi-separated schemes over T such that

1. P is invariant under thickenings;
2. if T' is the limit of a directed inverse system of quasi-compact quasi-separated schemes Tᵢ over T with affine transition morphisms and P(Tᵢ) holds for all i, then P(T') holds;
3. if P(T') holds, Z ⊂ T' is a closed subscheme and its complement U is quasi-compact, then P(U) or P(Z) holds;

and such that P(T) holds. Then there are a field K and a morphism Spec K ⟶ T with P(Spec K).

The limit hypothesis is the source's, for directed inverse systems; it is reduced to the one of AlgebraicGeometry.exists_field_point, for small cofiltered diagrams, by AlgebraicGeometry.limit_property_of_isCofiltered. The [Nonempty T] hypothesis is necessary, as recorded at AlgebraicGeometry.exists_field_point.

public theorem exists_field_point_of_isDirected {T : Scheme.{u}} [Nonempty T] [CompactSpace T]
    [QuasiSeparatedSpace T] (P : ∀ X : Scheme.{u}, (X ⟶ T) → Prop)
    (hThickening : ∀ {X X' : Scheme.{u}} [CompactSpace X] [QuasiSeparatedSpace X]
      [CompactSpace X'] [QuasiSeparatedSpace X'] (i : X ⟶ X') (f : X' ⟶ T),
      IsThickening i → (P X (i ≫ f) ↔ P X' f))
    (hLimit : ∀ {I : Type u} [Preorder I] [IsDirected I (· ≤ ·)] [Nonempty I]
      (D : Iᵒᵖ ⥤ Scheme.{u})
      [∀ {i j : Iᵒᵖ} (a : i ⟶ j), IsAffineHom (D.map a)]
      [∀ i, CompactSpace (D.obj i)] [∀ i, QuasiSeparatedSpace (D.obj i)]
      (c : Cone D) (_ : IsLimit c)
      [CompactSpace c.pt] [QuasiSeparatedSpace c.pt]
      (f : c.pt ⟶ T) (fi : ∀ i, D.obj i ⟶ T),
      (∀ {i j : Iᵒᵖ} (a : i ⟶ j), D.map a ≫ fi j = fi i) →
      (∀ i, c.π.app i ≫ fi i = f) →
      (∀ i, P (D.obj i) (fi i)) → P c.pt f)
    (hClosedOrOpen : ∀ {X : Scheme.{u}} [CompactSpace X]
      [QuasiSeparatedSpace X] (f : X ⟶ T) (J : X.IdealSheafData)
      [CompactSpace J.subscheme] [QuasiSeparatedSpace J.subscheme]
      (V : X.Opens) (_hV : (V : Set X) = (J.support : Set X)ᶜ)
      [CompactSpace V.toScheme] [QuasiSeparatedSpace V.toScheme],
      P X f → (P V.toScheme (V.ι ≫ f) ∨
        P J.subscheme (J.subschemeι ≫ f)))
    (hP : P T (𝟙 T)) :
    ∃ (K : Type u) (_ : Field K) (g : Spec (.of K) ⟶ T),
      P (Spec (.of K)) g

Tag 0F07, a lemma

Tag 0F07 in the Stacks Project.

Stacks.EtaleCohomology.exists_affine_field_not_isIso_higherBaseChangeMapAb, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/FirstFailureFieldObstruction.lean, line 238.

Stacks Project, Tag 0F07. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q) holds and BC(f, n, q + 1) does not. Then there are a commutative diagram

X  ←--  X'  ←-h-  Y
|f      |f'       |e
S  ←--  S'  ←-g-  Spec K

with X' = X ×_S S', Y = X' ×_{S'} Spec K, S' affine and K a field, and an abelian sheaf F on the small étale site of Spec K annihilated by n, such that (f')⁻¹ R^{q+1} g_* F ⟶ R^{q+1} h_* e⁻¹ F is not an isomorphism.

The source's "we may and do replace S' by an affine open of S'" is Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_forall_isAffine. The dévissage AlgebraicGeometry.exists_field_point_of_isDirected of [Stacks, Tag 0F06] is applied to the source's property P, which is Stacks.EtaleCohomology.ObstructionSurvives; its three hypotheses are Stacks.EtaleCohomology.obstructionSurvives_iff_of_isThickening, Stacks.EtaleCohomology.obstructionSurvives_of_isLimit and Stacks.EtaleCohomology.obstructionSurvives_open_or_closed.

Proved here: that the failure of BC(f, n, q + 1) under BC(f, n, q) is a failure in degree exactly q + 1, that the shrunk base is quasi-compact and quasi-separated, the choice of the obstruction class and that P holds at the base, that P at a field-valued point is a failure of base change, and that the sheaf the dévissage returns is annihilated by n.

public theorem exists_affine_field_not_isIso_higherBaseChangeMapAb {X S : Scheme.{u}} (f : X ⟶ S)
    (n : ℤ) (q : ℕ) (hprev : BaseChangeHolds f n (q : ℤ))
    (hfail : ¬ BaseChangeHolds f n ((q : ℤ) + 1)) :
    ∃ (S' X' Y : Scheme.{u}) (_ : IsAffine S') (s : S' ⟶ S) (p : X' ⟶ X) (f' : X' ⟶ S')
      (_hp : IsPullback p f' f s) (K : Type u) (_ : Field K) (g : Spec (.of K) ⟶ S')
      (h : Y ⟶ X') (e : Y ⟶ Spec (.of K)) (sq : IsPullback h e f' g)
      (F : Sheaf (smallEtaleTopology (Spec (.of K))) Ab.{u + 1}),
      (∀ (U : (Spec (.of K)).Etale) (x : F.obj.obj (op U)), n • x = 0) ∧
        ¬ IsIso ((higherBaseChangeMapAb f' g h e sq.w (q + 1)).app F)

Tag 0F08, a lemma

Tag 0F08 in the Stacks Project.

Stacks.EtaleCohomology.FirstFailureDiagram.normalFailureDiagram_of_not_baseChangeHolds, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/FirstFailureDiagram.lean, line 412. The attribute's comment: “the base change condition BC(f, n, q) of 0F02 and its failure in degree q + 1, indexed as the source's BC(f, n, q - 1) and BC(f, n, q)”.

Stacks Project, Tag 0F08. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q) holds and BC(f, n, q + 1) does not (the source's BC(f, n, q - 1) and BC(f, n, q) for q ≥ 1, indexed from q + 1). Then there are a commutative diagram

X  ←--  X'  ←-h-  Y
|f      |f'       |e
S  ←--  S'  ←-g-  Spec K

with both squares cartesian, in which S' is affine, integral and normal with algebraically closed function field, K is algebraically closed and g : Spec K ⟶ S' is dominant, and an integer d ∣ n for which R^{q+1} h_*(ℤ/dℤ) is nonzero; this is NormalFailureDiagram, the sheaf ℤ/dℤ on Y being the inverse image along e of constantZMod (Spec K) d, the constant sheaf ℤ/dℤ on Spec K. BaseChangeHolds is the condition BC(f, n, q₀) of [Stacks, Tag 0F02].

The failure is first realised over the spectrum of a field K with S' affine by [Stacks, Tag 0F07] (Stacks.EtaleCohomology.exists_affine_field_not_isIso_higherBaseChangeMapAb). The field is replaced by an algebraic closure K̄ by [Stacks, Tag 0F04] (Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_integralSurjective) applied to the integral surjective Spec K̄ ⟶ Spec K, and the diagram is then normalized by normalFailureDiagram_of_isAlgClosed.

public theorem normalFailureDiagram_of_not_baseChangeHolds {X S : Scheme.{u}} (f : X ⟶ S)
    (n : ℤ) (q : ℕ) (hprev : BaseChangeHolds f n (q : ℤ))
    (hfail : ¬ BaseChangeHolds f n ((q : ℤ) + 1)) :
    NormalFailureDiagram f n (q + 1)

Stacks.EtaleCohomology.FirstFailureDiagram.normalFailureDiagram_of_firstFailure, an alias in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/BaseChangeHigher/FirstFailureDiagramNormalization.lean, line 128.

With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume BC(f, n, q) holds and BC(f, n, q + 1) does not. Then there is a diagram

X  ←--  X'  ←-h-  Y
|f      |         |
S  ←--  S'  ←---  Spec K

with both squares cartesian, S' affine, integral and normal with algebraically closed function field, K algebraically closed with Spec K ⟶ S' dominant, and an integer d ∣ n with R^{q+1} h_* (ℤ/dℤ) nonzero, ℤ/dℤ on Y being the inverse image of the constant sheaf on Spec K (Stacks.EtaleCohomology.FirstFailureDiagram.NormalFailureDiagram). This is another name for Stacks.EtaleCohomology.FirstFailureDiagram.normalFailureDiagram_of_not_baseChangeHolds.

public alias normalFailureDiagram_of_firstFailure

Sources

• Section 59.88, Tag 0F01
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.higherBaseChangeMapAb · Stacks.EtaleCohomology.BaseChangeHolds · Stacks.EtaleCohomology.mono_cokernelDesc_higherBaseChangeMapAb_app_of_injective · Stacks.EtaleCohomology.mono_higherBaseChangeMapAb_app_of_annihilated · Stacks.EtaleCohomology.mono_cokernelMap_higherBaseChangeMapAb_app_of_annihilated · Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_isIso_integralSurjective · Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_of_integralSurjective · Stacks.EtaleCohomology.isIso_higherBaseChangeMapAb_app_iff_of_isIntegralHom

Section59.89 Smooth base change#

Section 59.89 of the Stacks Project, Tag 0EYQ. 3 tags here are carried by 3 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0EYT, a lemma

Tag 0EYT in the Stacks Project.

Stacks.EtaleCohomology.exists_galoisCover_killing_class, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/CurveClassKilling.lean, line 366.

Stacks Project, Tag 0EYT. Let K/k be a field extension, X a smooth affine curve over k with a rational point x, F an abelian sheaf on Spec(K)_étale annihilated by an integer n invertible in k, q > 0 and ξ ∈ H^q(X_K, F). There are finite extensions K'/K and k'/k with k' ⊆ K' and a finite étale Galois cover Z ⟶ X_{k'}, whose group has order dividing a power of n and which is split over x_{k'}, such that ξ dies in H^q(Z_{K'}, F).

theorem exists_galoisCover_killing_class [IsCurve k f] [Smooth f] [IsAffine X]
    (x : Spec (.of k) ⟶ X) (hx : x ≫ f = 𝟙 _)
    (F : Sheaf (Spec (.of K)).smallEtaleTopology AddCommGrpCat.{u + 1}) {n : ℕ}
    (hn : IsUnit (n : k)) (hF : n • 𝟙 F = 0) {q : ℕ} (hq : 0 < q)
    (ξ : Sheaf.H.{u + 1, u + 1}
      ((etaleAbelianInverseImage.{u, u + 1} (pullback.fst (spectrumMap k K) f)).obj F) q) :
    ∃ (k' K' : Type u) (_ : Field k') (_ : Field K') (_ : Algebra k k') (_ : Algebra K K')
      (_ : Algebra k' K') (_ : Algebra k K') (_ : IsScalarTower k k' K')
      (_ : IsScalarTower k K K') (_ : FiniteDimensional k k') (_ : FiniteDimensional K K')
      (Z : Scheme.{u}) (p : Z ⟶ pullback (spectrumMap k k') f),
      IsGaloisCover p ∧ (∃ m : ℕ, Nat.card (Aut (Over.mk p)) ∣ n ^ m) ∧
        IsSplitOver p (basePoint f k' x hx) ∧ coverBaseChangeClass f K k' K' p F q ξ = 0

Tag 0EYU, a theorem

Tag 0EYU in the Stacks Project.

Stacks.EtaleCohomology.SmoothBaseChange.isIso_higherBaseChangeMapAb_app_of_smooth, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/Unconditional.lean, line 53.

The smooth base change theorem ([Stacks, Theorem 0EYU]). In a cartesian square

X  ←-h-  Y
|f       |e
S  ←-g-  T

with f smooth and g quasi-compact and quasi-separated, the base change map f⁻¹ R^q g_* F ⟶ R^q h_* e⁻¹ F is an isomorphism for every q and every abelian sheaf F on T_étale all of whose stalks at geometric points are torsion of orders invertible on S: every element of every such stalk is killed by a positive integer n which is a unit on S.

theorem isIso_higherBaseChangeMapAb_app_of_smooth
    {X Y S T : Scheme.{u}} (f : X ⟶ S) (h : Y ⟶ X) (e : Y ⟶ T) (g : T ⟶ S)
    (sq : IsPullback h e f g) [Smooth f] [QuasiCompact g] [QuasiSeparated g]
    (F : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (hF : ∀ (t : GeometricPoint T) (a : (smallEtaleStalkAb t.hom).obj F),
      ∃ n : ℕ, 0 < n ∧ IsUnit ((n : ℤ) : Γ(S, ⊤)) ∧ n • a = 0) (q : ℕ) :
    IsIso ((higherBaseChangeMapAb f g h e sq.w q).app F)

Tag 0F09, a lemma

Tag 0F09 in the Stacks Project.

Stacks.EtaleCohomology.isIso_smallEtaleDerivedBaseChangeAb_app_of_smoothLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/SmoothLimitBaseChangeStalks.lean, line 114.

Smooth base change along a limit of smooth schemes ([Stacks, Tag 0F09]). Let S be a scheme and let S' = lim S_i be a directed inverse limit of schemes S_i smooth over S with affine transition morphisms: an inverse system D over a directed nonempty preorder I, with structure morphisms p_i : S_i ⟶ S compatible with the transitions, a limit cone c with point S', and g : S' ⟶ S the composite of each leg of c with p_i. Let f : X ⟶ S be quasi-compact and quasi-separated and let X' = X ×_S S' be a fibre product, with projections f' : X' ⟶ S' and g' : X' ⟶ X. Then for every E ∈ D⁺(X_étale) whose cohomology sheaves H^q(E) have stalks which are torsion of orders invertible on S (every element of the stalk at every geometric point is killed by a positive integer which is a unit on S), the base change map g⁻¹ Rf_* E ⟶ Rf'_* (g')⁻¹ E is an isomorphism.

theorem isIso_smallEtaleDerivedBaseChangeAb_app_of_smoothLimit
    {I : Type u} [Preorder I] [IsDirected I (· ≤ ·)] [Nonempty I] {S : Scheme.{u}}
    (D : Iᵒᵖ ⥤ Scheme.{u}) (p : D ⟶ (Functor.const Iᵒᵖ).obj S) (hp : ∀ i, Smooth (p.app i))
    (haff : ∀ ⦃i j : Iᵒᵖ⦄ (a : i ⟶ j), IsAffineHom (D.map a)) (c : Cone D) (hc : IsLimit c)
    (g : c.pt ⟶ S) (hg : ∀ i, c.π.app i ≫ p.app i = g) {X X' : Scheme.{u}} (f : X ⟶ S)
    [QuasiCompact f] [QuasiSeparated f] (f' : X' ⟶ c.pt) (g' : X' ⟶ X)
    (sq : IsPullback f' g' g f) (E : DerivedCategory.Plus (Sheaf (smallEtaleTopology X) Ab.{u + 1}))
    (hE : ∀ (q : ℤ) (t : GeometricPoint X)
      (a : ↑((smallEtaleStalkAb t.hom).obj
        ((DerivedCategory.Plus.homologyFunctor
          (Sheaf (smallEtaleTopology X) Ab.{u + 1}) q).obj E))),
      ∃ n : ℕ, 0 < n ∧ IsUnit ((n : ℤ) : Γ(S, ⊤)) ∧ n • a = 0) :
    IsIso ((smallEtaleDerivedBaseChangeAb g f f' g' sq.w).app E)

Sources

• Section 59.89, Tag 0EYQ
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.exists_galoisCover_killing_class · Stacks.EtaleCohomology.SmoothBaseChange.isIso_higherBaseChangeMapAb_app_of_smooth · Stacks.EtaleCohomology.isIso_smallEtaleDerivedBaseChangeAb_app_of_smoothLimit

Section59.90 Applications of smooth base change#

Section 59.90 of the Stacks Project, Tag 0F0A. 3 tags here are carried by 4 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F1C, a lemma

Tag 0F1C in the Stacks Project.

Stacks.EtaleCohomology.FieldExtensionBaseChange.isIso_smallEtaleDerivedBaseChangeAb_of_stalks, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/FieldExtensionBaseChangeTheorem.lean, line 65. The attribute's comment: “the base change map (S_L ⟶ S)⁻¹ Rg_* E ⟶ Rg_{L,*} E_L is an isomorphism”.

Direct images commute with extension of the base field ([Stacks, Tag 0F1C]). Let L/K be an extension of fields, let g : T ⟶ S be a quasi-compact and quasi-separated morphism of schemes over K, and let g_L : T_L ⟶ S_L be its base change to Spec L, with S_L = Spec L ×_K S and T_L = Spec L ×_K T. Let E ∈ D⁺(T_étale) have cohomology sheaves whose stalks are torsion of orders invertible in K: every element of the stalk of H^q(E) at every geometric point of T is killed by a positive integer invertible in K. Let E_L be the pullback of E to (T_L)_étale. Then Rg_{L,*} E_L is the pullback of Rg_* E to S_L: the base change map (S_L ⟶ S)⁻¹ Rg_* E ⟶ Rg_{L,*} E_L is an isomorphism.

theorem isIso_smallEtaleDerivedBaseChangeAb_of_stalks
    (E : DerivedCategory.Plus (Sheaf (smallEtaleTopology T) Ab.{u + 1}))
    (hE : ∀ (q : ℤ) (t : GeometricPoint T)
      (a : ↑((smallEtaleStalkAb t.hom).obj ((DerivedCategory.Plus.homologyFunctor _ q).obj E))),
      ∃ n : ℕ, 0 < n ∧ IsUnit (n : K) ∧ n • a = 0) :
    IsIso ((smallEtaleDerivedBaseChangeAb (pullback.snd _ sS) g
      (baseChangeMap (Spec.map (CommRingCat.ofHom (algebraMap K L))) sS sT g hg)
      (pullback.snd _ sT) (baseChangeMap_snd _ sS sT g hg)).app E)

Tag 0F0B, a lemma

Tag 0F0B in the Stacks Project.

Stacks.EtaleCohomology.bijective_etaleCohomology_pullback_of_isSepClosed, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/SeparablyClosedFieldInvariance.lean, line 71. The attribute's comment: “(1): H^q(X, E) ⟶ H^q(X_K, E|_{X_K}) is an isomorphism for K/k separably closed”. References: SGA 4 XII (5.4), “special case”; SGA 4 XVI (1.6), “Abelian L-torsion clause”; SGA 4½ Arcata (5.3.3), “special case”; SGA 4½ Th finitude (3.9), “more general”.

[Stacks, Tag 0F0B] (1). Let K/k be an extension of separably closed fields, X a quasi-compact and quasi-separated scheme over k, p : X_K ⟶ X its base change to K and E in D⁺(X_étale) whose cohomology sheaves have stalks which are torsion of orders invertible in k. Then for every q the restriction map H^q_ét(X, E) ⟶ H^q_ét(X_K, E|_{X_K}) is bijective. Here H^q_ét(X, E) is the group of morphisms in D⁺(X_étale) from the constant sheaf ℤ placed in degree -q to E, and restriction is the derived pullback along p followed by the identification of the pullback of ℤ_X with ℤ_{X_K}.

theorem bijective_etaleCohomology_pullback_of_isSepClosed
    {k K : Type u} [Field k] [Field K] [Algebra k K] [IsSepClosed k] [IsSepClosed K]
    {X XK : Scheme.{u}} (f : X ⟶ Spec (.of k)) [QuasiCompact f] [QuasiSeparated f]
    (p : XK ⟶ X) (fK : XK ⟶ Spec (.of K))
    (sq : IsPullback p fK f (Spec.map (CommRingCat.ofHom (algebraMap k K))))
    (E : DerivedCategory.Plus (Sheaf (smallEtaleTopology X) Ab.{u + 1}))
    (hE : ∀ q : ℤ, HasTorsionOrdersInvertibleOn (Spec (.of k))
      ((DerivedCategory.Plus.homologyFunctor (Sheaf (smallEtaleTopology X) Ab.{u + 1}) q).obj E))
    (q : ℤ) :
    Function.Bijective
      (fun φ : (DerivedCategory.Plus.singleFunctor _ (-q)).obj (constantZ X) ⟶ E ↦
        (singleConstantZPullbackIso p (-q)).hom ≫
          (smallEtalePullbackAb p).rightDerivedFunctorPlus.map φ)

Stacks.EtaleCohomology.isIso_derivedUnit_baseChange_of_isSepClosed, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/SeparablyClosedFieldInvariance.lean, line 93. The attribute's comment: “(2): E ⟶ R(X_K → X)_* E|_{X_K} is an isomorphism for K/k separably closed”. References: SGA 4 XVI (1.6), “partial”.

[Stacks, Tag 0F0B] (2). Let K/k be an extension of separably closed fields, X a scheme over k, p : X_K ⟶ X its base change to K and E in D⁺(X_étale) whose cohomology sheaves have stalks which are torsion of orders invertible in k. Then E ⟶ Rp_*(E|_{X_K}), the unit of the derived adjunction p⁻¹ ⊣ Rp_* at E, is an isomorphism. The source assumes X quasi-compact and quasi-separated; the statement holds without it.

theorem isIso_derivedUnit_baseChange_of_isSepClosed
    {k K : Type u} [Field k] [Field K] [Algebra k K] [IsSepClosed k] [IsSepClosed K]
    {X XK : Scheme.{u}} (f : X ⟶ Spec (.of k)) (p : XK ⟶ X) (fK : XK ⟶ Spec (.of K))
    (sq : IsPullback p fK f (Spec.map (CommRingCat.ofHom (algebraMap k K))))
    (E : DerivedCategory.Plus (Sheaf (smallEtaleTopology X) Ab.{u + 1}))
    (hE : ∀ q : ℤ, HasTorsionOrdersInvertibleOn (Spec (.of k))
      ((DerivedCategory.Plus.homologyFunctor (Sheaf (smallEtaleTopology X) Ab.{u + 1}) q).obj E)) :
    IsIso ((smallEtaleAdjunctionAb p).rightDerivedFunctorPlus.unit.app E)

Tag 0F1D, a lemma

Tag 0F1D in the Stacks Project.

Stacks.EtaleCohomology.FirstFailureFunctionField.functionFieldFailure_of_not_baseChangeHolds, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/FirstFailureOverFunctionField.lean, line 78.

Stacks Project, Tag 0F1D. With f : X ⟶ S and n as in [Stacks, Tag 0F02], assume n is invertible on S, BC(f, n, q) holds and BC(f, n, q + 1) does not. Then there are cartesian squares

X  ←--  X'  ←-h-  Y
|f      |         |
S  ←--  S'  ←---  Spec K

where S' is affine, integral and normal with algebraically closed function field K = K(S'), the bottom right arrow is the canonical morphism from the spectrum of the function field, and there is an integer d ∣ n with R^{q+1} h_*(ℤ/dℤ) nonzero (Stacks.EtaleCohomology.FirstFailureFunctionField.FunctionFieldFailure).

public theorem functionFieldFailure_of_not_baseChangeHolds {X S : Scheme.{u}} (f : X ⟶ S)
    (n : ℤ) (q : ℕ) (hn : IsUnit (n : Γ(S, ⊤))) (hprev : BaseChangeHolds f n (q : ℤ))
    (hfail : ¬ BaseChangeHolds f n ((q : ℤ) + 1)) : FunctionFieldFailure f n (q + 1)

Sources

• Section 59.90, Tag 0F0A
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.FieldExtensionBaseChange.isIso_smallEtaleDerivedBaseChangeAb_of_stalks · Stacks.EtaleCohomology.bijective_etaleCohomology_pullback_of_isSepClosed · Stacks.EtaleCohomology.isIso_derivedUnit_baseChange_of_isSepClosed · Stacks.EtaleCohomology.FirstFailureFunctionField.functionFieldFailure_of_not_baseChangeHolds

Section59.91 The proper base change theorem#

Section 59.91 of the Stacks Project, Tag 095S. 15 tags here are carried by 19 declarations of the repository (all in StacksFrontier) and 4 registered declarations, quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0A0B, a lemma

Tag 0A0B in the Stacks Project.

AlgebraicGeometry.bijective_globalSectionsRestriction_closedFibre_of_henselianPair_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ClosedFibreGlobalSections.lean, line 223.

For (A, I) a henselian pair and f : X ⟶ Spec A proper, restriction to the closed fibre X ×_{Spec A} Spec (A ⧸ I) is bijective on global sections of every sheaf of sets on the space of X.

theorem bijective_globalSectionsRestriction_closedFibre_of_henselianPair_of_isProper
    (A : Type u) [CommRing A] (I : Ideal A) (hI : HenselianPair A I) {X : Scheme.{u}}
    (f : X ⟶ Spec (.of A)) [IsProper f]
    (F : TopCat.Sheaf (Type u) (TopCat.of X)) :
    Function.Bijective (Stacks.globalSectionsRestriction
      (Set.range (pullback.fst f (Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I))))) F)

Tag 0A0C, a lemma

Tag 0A0C in the Stacks Project.

Stacks.EtaleCohomology.isIso_globalSectionsComparison_closedFibre_of_henselianPair_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ClosedFibreEtaleSectionsProper.lean, line 119. References: SGA 4 XII (6.13), “partial, special case”.

[Stacks, Tag 0A0C]. Let (A, I) be a henselian pair and let f : X ⟶ Spec A be proper. For every sheaf of sets ℱ on the small étale site of X, restriction of global sections to the closed fibre Z = X ×_{Spec A} Spec (A ⧸ I) is bijective: the canonical comparison Γ(X, ℱ) ⟶ Γ(Z, i_{small}^{-1}ℱ), natural in ℱ, is an isomorphism.

This is [Stacks, Tag 09ZF] for the closed immersion i : Z ⟶ X, whose two hypotheses are [Stacks, Tag 0A0B] for f and for Y ⟶ X ⟶ Spec A, Y ⟶ X finite.

public theorem isIso_globalSectionsComparison_closedFibre_of_henselianPair_of_isProper :
    IsIso (globalSectionsSmallEtaleInverseImageComparison.{u, v}
      (pullback.fst f (Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I))))).val

Stacks.EtaleCohomology.bijective_globalSectionsComparison_app_closedFibre_of_henselianPair_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ClosedFibreEtaleSectionsProper.lean, line 137. References: SGA 4 XII (6.13), “partial, special case”.

[Stacks, Tag 0A0C], sheaf by sheaf. Let (A, I) be a henselian pair and let f : X ⟶ Spec A be proper. For every sheaf of sets ℱ on the small étale site of X, the restriction map Γ(X, ℱ) ⟶ Γ(Z, i_{small}^{-1}ℱ) to the closed fibre Z = X ×_{Spec A} Spec (A ⧸ I) is bijective. This is the component at ℱ of isIso_globalSectionsComparison_closedFibre_of_henselianPair_of_isProper.

public theorem bijective_globalSectionsComparison_app_closedFibre_of_henselianPair_of_isProper
    (F : Sheaf (Scheme.smallEtaleTopology X) (Type (max (u + 1) v))) :
    Function.Bijective ((globalSectionsSmallEtaleInverseImageComparison.{u, v}
      (pullback.fst f (Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I))))).val.app F)

Tag 0A3S, a lemma

Tag 0A3S in the Stacks Project.

Stacks.EtaleCohomology.isIso_globalSectionsComparison_closedFibre_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ClosedFibreEtaleSectionsProper.lean, line 152. References: SGA 4 XII (6.13), “special case”.

[Stacks, Tag 0A3S]. Let A be a henselian local ring, let f : X ⟶ Spec A be proper and let X₀ = X ×_{Spec A} Spec κ(A) be the fibre of f over the closed point. For every sheaf of sets ℱ on the small étale site of X, Γ(X, ℱ) = Γ(X₀, ℱ|_{X₀}): the canonical comparison of global sections along the inclusion of X₀, natural in ℱ, is an isomorphism.

This is [Stacks, Tag 0A0C] for the henselian pair (A, 𝔪).

public theorem isIso_globalSectionsComparison_closedFibre_of_isProper [HenselianLocalRing A] :
    IsIso (globalSectionsSmallEtaleInverseImageComparison.{u, v}
      (closedFibreInclusion A f)).val

Stacks.EtaleCohomology.bijective_globalSectionsComparison_app_closedFibre_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ClosedFibreEtaleSectionsProper.lean, line 165.

[Stacks, Tag 0A3S], sheaf by sheaf. Let A be a henselian local ring, let f : X ⟶ Spec A be proper and let X₀ = X ×_{Spec A} Spec κ(A) be the fibre over the closed point. For every sheaf of sets ℱ on the small étale site of X, restriction of global sections Γ(X, ℱ) ⟶ Γ(X₀, ℱ|_{X₀}) is bijective. This is the component at ℱ of isIso_globalSectionsComparison_closedFibre_of_isProper.

public theorem bijective_globalSectionsComparison_app_closedFibre_of_isProper
    [HenselianLocalRing A] (F : Sheaf (Scheme.smallEtaleTopology X) (Type (max (u + 1) v))) :
    Function.Bijective ((globalSectionsSmallEtaleInverseImageComparison.{u, v}
      (closedFibreInclusion A f)).val.app F)

Tag 0A3T, a lemma

Tag 0A3T in the Stacks Project.

Stacks.EtaleCohomology.pushforwardSheafFiberClosedPointIsoGeometricFibreGamma, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/GeometricFibreSections.lean, line 494. The attribute's comment: “conditional on the base being Spec R for a strictly henselian local ring R, the reduction of a general base to the strict henselization being Tag 03Q9 for sheaves of sets, which is absent; and carrying hcompat, the hypothesis of the library's Tag 0A3S”.

The stalk of a direct image along a proper morphism over a strictly henselian local base is the set of global sections over the geometric fibre, granted compatible presenting families. For R strictly henselian local with residue field κ(R), f : X ⟶ Spec R proper, Ω an extension of κ(R) and x a geometric point of Spec R above the closed point, (f_*F)_x = Γ(X_Ω, F|_{X_Ω}) with X_Ω = (X ×_{Spec R} Spec κ(R)) ×_{Spec κ(R)} Spec Ω, naturally in the sheaf of sets F.

This is the whole of the proof of [Stacks, Tag 0A3T] except for the reduction of a general base to the strict henselization, which is Tag 03Q9 and is absent: Tag 03Q9 over the already strictly henselian base is pushforwardSheafFiberClosedPointIsoGamma, Tag 0A3S is the hypothesis hcompat, and the extension of the separably closed residue field to Ω is [Stacks, Tag 0A3I]. That the right-hand side is the geometric fibre of f at x is closedFibreBaseChange_fst_comp_closedFibreInclusion, for x the composite point.

def pushforwardSheafFiberClosedPointIsoGeometricFibreGamma [IsProper f]
    (hcompat : ∀ (F : Sheaf (Scheme.smallEtaleTopology X) (Type (max (u + 1) v)))
      (t : ((smallEtaleInverseImage.{u, v} (closedFibreInclusion R f)).obj F).obj.obj
        (op (terminal (closedFibre R f).Etale))),
      ∃ (α : Type (u + 1)) (W : α → X.Etale) (s : ∀ j, F.obj.obj (op (W j))),
        (⨆ j, smallEtaleImage (W j)) = ⊤ ∧ SmallEtaleCompatible F W s ∧
          ∀ j, ((smallEtaleInverseImage.{u, v} (closedFibreInclusion R f)).obj F).obj.map
                (terminal.from ((smallEtaleBaseChange
                  (closedFibreInclusion R f)).obj (W j))).op t
              = smallEtaleSectionPullback.{u, v} (closedFibreInclusion R f) F (W j) (s j)) :
    (smallEtaleBaseChange f).sheafPushforwardContinuous (Type (max (u + 1) v))
          (smallEtaleTopology (Spec (.of R))) (smallEtaleTopology X) ⋙
        (pointSmallEtale x).sheafFiber ≅
      smallEtaleInverseImage.{u, v}
          (pullback.fst (closedFibreProjection R f)
            (Spec.map (CommRingCat.ofHom
              (algebraMap (IsLocalRing.ResidueField R) Ω))) ≫ closedFibreInclusion R f) ⋙
        Sheaf.Γ (smallEtaleTopology (pullback (closedFibreProjection R f)
          (Spec.map (CommRingCat.ofHom (algebraMap (IsLocalRing.ResidueField R) Ω)))))
          (Type (max (u + 1) v))

Declarations registered for the tag:

• Stacks.EtaleCohomology.isIso_properGeometricFibreComparison_of_isProper, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperBaseChangeStacksTags.lean, line 47 (StacksFrontier).

Tag 0A3U, a lemma

Tag 0A3U in the Stacks Project.

Declarations registered for the tag:

• Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isProper, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperBaseChangeStacksTags.lean, line 49 (StacksFrontier).

Tag 0A2A, an equation

Tag 0A2A in the Stacks Project.

Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/BaseChangeMap.lean, line 158.

The canonical base change map g^{-1} Rf_* ⟶ Rf'_* (g')^{-1} on the unbounded derived categories of abelian sheaves for the small étale topology, attached to a commutative square g' ≫ f = f' ≫ g of schemes.

public def etaleAbelianDerivedBaseChange :
    etaleAbelianDerivedDirectImage.{u, v} f ⋙ etaleAbelianDerivedInverseImage.{u, v} g ⟶
      etaleAbelianDerivedInverseImage.{u, v} g' ⋙ etaleAbelianDerivedDirectImage.{u, v} f'

Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange_fac, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/BaseChangeMap.lean, line 241. The attribute's comment: “canonical comparison on unbounded complexes”.

For every complex K of abelian sheaves on X_étale, the base change map at the object it represents, precomposed with the exactness comparison for g^{-1} and with the derived-functor unit of Rf_*, is the degreewise base change comparison followed by the derived-functor unit of Rf'_*.

public theorem etaleAbelianDerivedBaseChange_fac :
    (etaleAbelianInverseImage.{u, v} g).mapDerivedCategoryFactorsh.inv.app
        (((etaleAbelianDirectImage.{u, v} f).mapHomotopyCategory
          (ComplexShape.up ℤ)).obj K) ≫
      (etaleAbelianDerivedInverseImage.{u, v} g).map
        ((etaleAbelianDerivedDirectImageUnit.{u, v} f).app K) ≫
      (etaleAbelianDerivedBaseChange.{u, v} f g g' f' h).app (DerivedCategory.Qh.obj K) =
    DerivedCategory.Qh.map ((etaleAbelianHomotopyBaseChange.{u, v} f g g' f' h).app K) ≫
      (etaleAbelianDerivedDirectImageUnit.{u, v} f').app
        (((etaleAbelianInverseImage.{u, v} g').mapHomotopyCategory
          (ComplexShape.up ℤ)).obj K) ≫
      (etaleAbelianDerivedDirectImage.{u, v} f').map
        ((etaleAbelianInverseImage.{u, v} g').mapDerivedCategoryFactorsh.inv.app K)

Tag 0A4A, an equation

Tag 0A4A in the Stacks Project.

Stacks.EtaleCohomology.etaleAbelianSheafDerivedBaseChange, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/BaseChangeMap.lean, line 277.

The base change map for a single abelian sheaf F on X_étale, placed in degree zero: the canonical morphism g^{-1} Rf_* F ⟶ Rf'_* (g')^{-1} F in the derived category of Y'_étale, attached to a commutative square g' ≫ f = f' ≫ g of schemes. It is the component of Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange at F[0].

public def etaleAbelianSheafDerivedBaseChange
    (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}) :
    (etaleAbelianDerivedInverseImage.{u, v} g).obj
        ((etaleAbelianDerivedDirectImage.{u, v} f).obj
          ((DerivedCategory.singleFunctor
            (Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}) 0).obj F)) ⟶
      (etaleAbelianDerivedDirectImage.{u, v} f').obj
        ((etaleAbelianDerivedInverseImage.{u, v} g').obj
          ((DerivedCategory.singleFunctor
            (Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}) 0).obj F))

Tag 0A4B, a lemma

Tag 0A4B in the Stacks Project.

Declarations registered for the tag:

• Stacks.EtaleCohomology.cohomologyCommutesWithBaseChange_iff_higherDirectImagesVanishOnInjectives, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperPushforwardBaseChangeRegistration.lean, line 15 (StacksFrontier): “(1) cohomology commutes with base change for f iff (2) higher direct images of pulled-back injective Z/lZ-sheaves vanish”.

Tag 0A4C, a lemma

Tag 0A4C in the Stacks Project.

Stacks.EtaleCohomology.isIso_etaleAbelianBaseChange_of_mono_unit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/SurjectiveCancellation.lean, line 297. The attribute's comment: “RESTRICTED: degree zero only — each cohomology commutes with base change is invertibility of the underived comparison of Tag 0A2A at every abelian sheaf, not of the derived comparison at every torsion sheaf, the bridge between the two being Tag 0A4B, absent here; hypothesis (3) is carried as monomorphy of the unit A ⟶ f_*f⁻¹A, which mono_etaleAbelianUnit_of_surjective supplies from Surjective f; properness is unused”.

Stacks 0A4C in degree zero: cancelling a surjection from a composite. Let f : X ⟶ Y and g : Y ⟶ Z be morphisms of schemes with named composite gf : X ⟶ Z, let c : Z' ⟶ Z, and consider commutative squares

X' --e--> X      Y' --d--> Y
|f'       |f     |g'       |g
v         v      v         v
Y' --d--> Y      Z' --c--> Z

with q : X' ⟶ Z' the named composite of f' and g'. Assume

1. the comparison d^{-1} f_* ⟶ f'_* e^{-1} of the upper square is invertible at every abelian sheaf on X_étale;
2. the comparison c^{-1} (gf)_* ⟶ q_* e^{-1} of the rectangle is invertible at every abelian sheaf on X_étale;
3. the unit A ⟶ f_* f^{-1} A is a monomorphism for every abelian sheaf A on Y_étale.

Then the comparison c^{-1} g_* ⟶ g'_* d^{-1} of the lower square is invertible at every abelian sheaf on Y_étale.

Hypothesis 3 is what surjectivity of f provides: Stacks.EtaleCohomology.mono_etaleAbelianUnit_of_surjective proves it from Surjective f at the coefficients Ab.{u + 1}. Properness of f and g is not used.

public theorem isIso_etaleAbelianBaseChange_of_mono_unit
    (f : X ⟶ Y) (g : Y ⟶ Z) (gf : X ⟶ Z) (hgf : gf = f ≫ g)
    (c : Z' ⟶ Z) (d : Y' ⟶ Y) (g' : Y' ⟶ Z') (e : X' ⟶ X) (f' : X' ⟶ Y')
    (q : X' ⟶ Z') (hq : q = f' ≫ g')
    (sq₁ : d ≫ g = g' ≫ c) (sq₂ : e ≫ f = f' ≫ d) (hyp : e ≫ gf = q ≫ c)
    (hunit : ∀ A : Sheaf Y.smallEtaleTopology AddCommGrpCat.{max (u + 1) v},
      Mono ((etaleAbelianInverseImageDirectImageAdj.{u, v} f).unit.app A))
    (h₁ : ∀ F : Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v},
      IsIso ((etaleAbelianBaseChange.{u, v} f d e f' sq₂).app F))
    (h₂ : ∀ F : Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v},
      IsIso ((etaleAbelianBaseChange.{u, v} gf c e q hyp).app F))
    (A : Sheaf Y.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}) :
    IsIso ((etaleAbelianBaseChange.{u, v} g c d g' sq₁).app A)

Stacks.EtaleCohomology.CohomologyCommutesWithBaseChange.of_comp_of_surjective, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/SurjectiveCancellationUnconditional.lean, line 233.

[Stacks, Tag 0A4C]: cancelling a surjection from a composite. Let f : X ⟶ Y and g : Y ⟶ Z be morphisms of schemes. Assume

1. cohomology commutes with base change for f,
2. cohomology commutes with base change for g ∘ f, and
3. f is surjective.

Then cohomology commutes with base change for g.

The source assumes f and g proper; only surjectivity of f and quasi-compactness and quasi-separatedness of g are used, so the statement is proved under those weaker hypotheses (a proper morphism is quasi-compact and quasi-separated). Cohomology commuting with base change is Stacks.EtaleCohomology.CohomologyCommutesWithBaseChange: the derived base change map c⁻¹ Rg_* F ⟶ Rg'_* d⁻¹ F is an isomorphism for every cartesian square and every torsion abelian sheaf F, in all degrees at once.

public theorem CohomologyCommutesWithBaseChange.of_comp_of_surjective (f : X ⟶ Y) (g : Y ⟶ Z)
    [Surjective f] [QuasiCompact g] [QuasiSeparated g]
    (hf : CohomologyCommutesWithBaseChange.{u, v} f)
    (hgf : CohomologyCommutesWithBaseChange.{u, v} (f ≫ g)) :
    CohomologyCommutesWithBaseChange.{u, v} g

Tag 0A4D, a lemma

Tag 0A4D in the Stacks Project.

Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange_isIso_of_compDirectImage, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/CompositeMorphismBaseChange.lean, line 328. The attribute's comment: “the derived half: for K in D(X_étale), if c⁻¹Rg_(Rf_K) → Rg'_d⁻¹(Rf_K) and d⁻¹Rf_K → Rf'_e⁻¹K are isomorphisms then c⁻¹R(g∘f)_K → R(g'∘f')_e⁻¹K is an isomorphism; the R^q statement of the same tag, and the reduction to injective Z/l-modules of 0A4B that the source proof uses to obtain it, are not covered here”.

Stacks 0A4D, the derived half. Let f : X ⟶ Y and g : Y ⟶ Z be morphisms of schemes with named composite gf : X ⟶ Z, let c : Z' ⟶ Z, and consider commutative squares

X' --e--> X      Y' --d--> Y
|f'       |f     |g'       |g
v         v      v         v
Y' --d--> Y      Z' --c--> Z

with q : X' ⟶ Z' the named composite of f' and g'. Let K be an object of D(X_étale). If the base change map c⁻¹ Rg_* ⟶ Rg'_* d⁻¹ of the lower square is an isomorphism at Rf_* K, and the base change map d⁻¹ Rf_* ⟶ Rf'_* e⁻¹ of the upper square is an isomorphism at K, then the base change map c⁻¹ R(gf)_* K ⟶ Rq_* e⁻¹ K of the outer rectangle is an isomorphism.

This is the composition statement of the source in the derived formulation of Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange: both hypotheses and conclusion say that a base change map of unbounded derived categories is invertible, so no torsion condition is imposed on the coefficients. It is formal, by Stacks.EtaleCohomology.etaleAbelianDerivedBaseChange_compDirectImage_app, which exhibits the map of the rectangle as the composite of the two given maps with two isomorphisms; neither square need be cartesian and neither morphism need be proper.

public theorem etaleAbelianDerivedBaseChange_isIso_of_compDirectImage
    (f : X ⟶ Y) (g : Y ⟶ Z) (gf : X ⟶ Z) (hgf : gf = f ≫ g)
    (c : Z' ⟶ Z) (d : Y' ⟶ Y) (g' : Y' ⟶ Z') (e : X' ⟶ X) (f' : X' ⟶ Y')
    (q : X' ⟶ Z') (hq : q = f' ≫ g')
    (sq₁ : d ≫ g = g' ≫ c) (sq₂ : e ≫ f = f' ≫ d) (hyp : e ≫ gf = q ≫ c)
    (K : DerivedCategory (Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v}))
    (h₁ : IsIso ((etaleAbelianDerivedBaseChange.{u, v} g c d g' sq₁).app
      ((etaleAbelianDerivedDirectImage.{u, v} f).obj K)))
    (h₂ : IsIso ((etaleAbelianDerivedBaseChange.{u, v} f d e f' sq₂).app K)) :
    IsIso ((etaleAbelianDerivedBaseChange.{u, v} gf c e q hyp).app K)

Stacks.EtaleCohomology.CohomologyCommutesWithBaseChange.comp, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/CompositeMorphismBaseChangeUnconditional.lean, line 270.

[Stacks, Tag 0A4D]: cohomology commutes with base change for a composite. Let f : X ⟶ Y and g : Y ⟶ Z be proper morphisms of schemes. Assume

1. cohomology commutes with base change for f, and
2. cohomology commutes with base change for g.

Then cohomology commutes with base change for g ∘ f.

Cohomology commuting with base change is Stacks.EtaleCohomology.CohomologyCommutesWithBaseChange: the derived base change map c⁻¹ R(gf)_* F ⟶ Rq_* e⁻¹ F is an isomorphism for every cartesian square over g ∘ f and every torsion abelian sheaf F on X_étale, in all degrees at once. The proof follows the source: at an injective sheaf of ℤ/nℤ-modules the composition formula for base change maps reduces the statement to the conditions for f and g (isIso_etaleAbelianSheafDerivedBaseChange_comp_toSheaf_injective); dimension shifting carries it to every sheaf killed by n, which is how the source uses Tag 0A4B; and a torsion sheaf is the filtered colimit of its subsheaves killed by an integer, with which both sides commute as g ∘ f is quasi-compact and quasi-separated (Tag 09Z1).

public theorem CohomologyCommutesWithBaseChange.comp {f : X ⟶ Y} {g : Y ⟶ Z}
    [IsProper f] [IsProper g]
    (hf : CohomologyCommutesWithBaseChange.{u, v} f)
    (hg : CohomologyCommutesWithBaseChange.{u, v} g) :
    CohomologyCommutesWithBaseChange.{u, v} (f ≫ g)

Tag 0A4E, a lemma

Tag 0A4E in the Stacks Project.

Stacks.EtaleCohomology.isIso_etaleAbelianSheafDerivedBaseChange_of_isFinite, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/AbelianCoefficientTransfer.lean, line 618. The attribute's comment: “RESTRICTED: the cartesian square is carried as four objects and four morphisms of the underlying category of a given big étale site B, the conclusion being about their images under B.inclusion; that containment is the only hypothesis beyond the square being cartesian with f finite, and coefficients are the abelian groups of AddCommGrpCat.{u+1}, which is Ab.{u+1}. No torsion condition and no invertibility is assumed.”. References: SGA 4 XII (5.1), “(iii)”.

Stacks 0A4E: cohomology commutes with base change for a finite morphism. For a cartesian square

X' --g'--> X
|f'        |f
v          v
Y' --g---> Y

of the underlying category of a big étale site with f finite, and every abelian sheaf F on the small étale site of X, the base change map g^{-1} Rf_* F ⟶ Rf'_* (g')^{-1} F on unbounded derived categories is an isomorphism.

No torsion condition on F is imposed — the source states the lemma for torsion sheaves, and this is the stronger statement — and finiteness of f' is not assumed but obtained from the square. The two inputs are Stacks.EtaleCohomology.isIso_etaleAbelianSheafDerivedBaseChange_of_isFinite_of_isIso, which reduces the derived statement to the underived comparison using the vanishing of the positive higher direct images of a finite morphism, and Stacks.EtaleCohomology.isIso_etaleAbelianBaseChange_of_isFinite, which is Stacks 0959 at abelian coefficients.

public theorem isIso_etaleAbelianSheafDerivedBaseChange_of_isFinite
    (sq : IsPullback (B.inclusion.map g') (B.inclusion.map f') (B.inclusion.map f)
      (B.inclusion.map g)) [IsFinite (B.inclusion.map f)]
    (F : Sheaf (Scheme.smallEtaleTopology (B.inclusion.obj X)) AddCommGrpCat.{u + 1}) :
    IsIso (etaleAbelianSheafDerivedBaseChange.{u, 0} (B.inclusion.map f) (B.inclusion.map g)
      (B.inclusion.map g') (B.inclusion.map f') sq.w F)

Stacks.EtaleCohomology.cohomologyCommutesWithBaseChange_of_isFinite_univ, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/FiniteBaseChangeUniverse.lean, line 187. References: SGA 4 XII (5.2), “(iii), special case”; SGA 4½ Th finitude (5.1.3), “1.3.1, special case”.

Proper base change for finite morphisms. Cohomology commutes with base change for a finite morphism of schemes f : X ⟶ Y: for every cartesian square with X' = Y' ×_Y X and every torsion abelian sheaf F on X_étale, with values in AddCommGrpCat.{max (u + 1) v}, the base change map g⁻¹ Rf_* F ⟶ Rf'_* (g')⁻¹ F is an isomorphism. The torsion hypothesis is not used: isIso_etaleAbelianSheafDerivedBaseChange_of_isFinite_univ holds at every abelian sheaf.

theorem cohomologyCommutesWithBaseChange_of_isFinite_univ {X Y : Scheme.{u}} (f : X ⟶ Y)
    [IsFinite f] :
    CohomologyCommutesWithBaseChange.{u, v} f

Tag 0A4F, a lemma

Tag 0A4F in the Stacks Project.

Stacks.EtaleCohomology.cohomologyCommutesWithBaseChange_of_projectiveLine_unconditional, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProjectiveLineReductionGeneral.lean, line 197.

[Stacks, Tag 0A4F]. To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it for the morphism ℙ¹_S ⟶ S for every scheme S: if cohomology commutes with base change for every ℙ¹_S ⟶ S, then it does for every proper f.

public theorem cohomologyCommutesWithBaseChange_of_projectiveLine_unconditional
    (hline : ∀ S : Scheme.{u},
      CohomologyCommutesWithBaseChange.{u, v} (ProjectiveSpace 1 S ↘ S))
    {X Y : Scheme.{u}} (f : X ⟶ Y) (hf : IsProper f) :
    CohomologyCommutesWithBaseChange.{u, v} f

Tag 095T, a theorem

Tag 095T in the Stacks Project.

Declarations registered for the tag:

• Stacks.EtaleCohomology.isIso_etaleAbelianSheafDerivedBaseChange_of_isProper, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperBaseChangeStacksTags.lean, line 53 (StacksFrontier).

Tag 0DDE, a lemma

Tag 0DDE in the Stacks Project.

Stacks.EtaleCohomology.isIso_etaleAbelianDerivedBaseChange_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/DerivedProperBaseChange.lean, line 45. References: SGA 4 XVII (4.3.2), “special case”; SGA 4 XVII (5.2.7), “(ii), special case”.

Stacks Project, Tag 0DDE. Let f : X ⟶ Y be a proper morphism of schemes, g : Y' ⟶ Y a morphism of schemes, and X' = Y' ×_Y X, given as a cartesian square sq with projections f' : X' ⟶ Y' and g' : X' ⟶ X. Let E be an object of the derived category of abelian sheaves on X_étale which is bounded below and has torsion cohomology sheaves. Then the base change map g^{-1} Rf_* E ⟶ Rf'_* (g')^{-1} E of Tag 0A2A is an isomorphism.

theorem isIso_etaleAbelianDerivedBaseChange_of_isProper {X Y X' Y' : Scheme.{u}} (f : X ⟶ Y)
    [IsProper f] {g : Y' ⟶ Y} {g' : X' ⟶ X} {f' : X' ⟶ Y'} (sq : IsPullback g' f' f g)
    (E : DerivedCategory (Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})) (n : ℤ)
    (hE : E.IsGE n)
    (hT : Stacks.Torsion.torsionCohomologyProperty.{u + 1} X.smallEtaleTopology E) :
    IsIso ((etaleAbelianDerivedBaseChange.{u, 0} f g g' f' sq.w).app E)

Tag 0DDF, a lemma

Tag 0DDF in the Stacks Project.

Stacks.EtaleCohomology.etaleAbelianSheafFiberRightDerivedDirectImageIsoOfIsProper, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperGeometricFibreStalk.lean, line 48. The attribute's comment: “part (1)”.

Stacks 0DDF (1). For a proper morphism f : X ⟶ Y, a geometric point ȳ : Spec Ω ⟶ Y and a torsion abelian sheaf F on X_étale, the stalk of Rⁿf_* F at ȳ is the étale cohomology Hⁿ(X_ȳ, F_ȳ) of the restriction of F to the geometric fibre X_ȳ = X ×_Y Spec Ω.

def etaleAbelianSheafFiberRightDerivedDirectImageIsoOfIsProper
    (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})
    (hF : Stacks.Torsion.torsionSheafProperty X.smallEtaleTopology F) (n : ℕ) :
    letI := hasExt_sheaf_smallEtale_succ (pullback f y)
    (((pointSmallEtale y).ulift.{u + 1}).sheafFiber (A := AddCommGrpCat.{u + 1})).obj
      (((etaleAbelianDirectImage.{u, 0} f).rightDerived n).obj F) ≅
    AddCommGrpCat.of
      (Sheaf.H.{u + 1} ((etaleAbelianInverseImage.{u, 0} (pullback.fst f y)).obj F) n)

Stacks.EtaleCohomology.etaleAbelianSheafFiberHomologyDerivedDirectImageIsoOfIsProper, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperGeometricFibreStalk.lean, line 65. The attribute's comment: “part (2)”.

Stacks 0DDF (2). For a proper morphism f : X ⟶ Y, a geometric point ȳ : Spec Ω ⟶ Y and an object E of D⁺(X_étale) with torsion cohomology sheaves, the stalk at ȳ of the n-th cohomology sheaf of Rf_* E is the n-th hypercohomology of the restriction of E to the geometric fibre X_ȳ = X ×_Y Spec Ω.

def etaleAbelianSheafFiberHomologyDerivedDirectImageIsoOfIsProper
    (E : DerivedCategory (Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})) (m : ℤ)
    (hE : E.IsGE m)
    (hT : Stacks.Torsion.torsionCohomologyProperty.{u + 1} X.smallEtaleTopology E) (n : ℤ) :
    letI := isGrothendieckAbelian_sheaf_smallEtale_objectUniverse (pullback f y)
    (((pointSmallEtale y).ulift.{u + 1}).sheafFiber (A := AddCommGrpCat.{u + 1})).obj
      ((DerivedCategory.homologyFunctor (Sheaf Y.smallEtaleTopology AddCommGrpCat.{u + 1})
        n).obj ((etaleAbelianDerivedDirectImage.{u, 0} f).obj E)) ≅
    (DerivedCategory.homologyFunctor AddCommGrpCat.{u + 1} n).obj
      ((Stacks.DerivedFunctor.totalRightDerived
        (Sheaf.Γ (pullback f y).smallEtaleTopology AddCommGrpCat.{u + 1})).obj
        ((etaleAbelianDerivedInverseImage.{u, 0} (pullback.fst f y)).obj E))

Sources

• Section 59.91, Tag 095S
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.bijective_globalSectionsRestriction_closedFibre_of_henselianPair_of_isProper · Stacks.EtaleCohomology.isIso_globalSectionsComparison_closedFibre_of_henselianPair_of_isProper · Stacks.EtaleCohomology.bijective_globalSectionsComparison_app_closedFibre_of_henselianPair_of_isProper · Stacks.EtaleCohomology.isIso_globalSectionsComparison_closedFibre_of_isProper · Stacks.EtaleCohomology.bijective_globalSectionsComparison_app_closedFibre_of_isProper · Stacks.EtaleCohomology.pushforwardSheafFiberClosedPointIsoGeometricFibreGamma · Stacks.EtaleCohomology.isIso_properGeometricFibreComparison_of_isProper · Stacks.EtaleCohomology.isIso_smallEtaleSetBaseChange_of_isProper

Section59.92 Applications of proper base change#

Section 59.92 of the Stacks Project, Tag 0A5I. 5 tags here are carried by 7 declarations of the repository (all in StacksFrontier) and 1 registered declaration, quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0DDG, a lemma

Tag 0DDG in the Stacks Project.

Stacks.EtaleCohomology.bijective_etaleCohomologyRestrictionMap_scalarExtensionFst_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/SeparablyClosedFieldExtension.lean, line 194.

Stacks Project, Tag 0DDG. Let K/k be an extension of separably closed fields, X a proper scheme over k (f : X ⟶ Spec k proper) and F a torsion abelian sheaf on X_étale. Then the restriction map H^q_étale(X, F) ⟶ H^q_étale(X_K, F|_{X_K}) along the projection X_K = X ×_{Spec k} Spec K ⟶ X is bijective for every q ≥ 0, with Ext computed in any universe w.

theorem bijective_etaleCohomologyRestrictionMap_scalarExtensionFst_of_isProper
    [HasExt.{w} (Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})]
    [HasExt.{w} (Sheaf (scalarExtension k K f).smallEtaleTopology AddCommGrpCat.{u + 1})]
    (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})
    (hF : Stacks.Torsion.torsionSheafProperty X.smallEtaleTopology F) (q : ℕ) :
    Function.Bijective
      (etaleCohomologyRestrictionMap.{u, 0, w} (scalarExtensionFst k K f) F q)

Tag 095U, a lemma

Tag 095U in the Stacks Project.

Stacks.EtaleCohomology.isZero_rightDerived_etaleModuleDirectImage_of_isIntegralHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/IntegralCohomologicalDimension.lean, line 132. The attribute's comment: “the case n = 0: for an integral morphism — a finite one, for instance, which is what a proper morphism with zero-dimensional fibres is — R^q f_* F = 0 for every q > 0 and every sheaf of Λ-modules F, with no torsion hypothesis. The general bound 2n is NOT proved: its induction needs the curve case, Tag 03SC, which is not formalized here.”. References: SGA 4 VIII (5.6), “more general”; SGA 4 Vbis (4.3.4), “more general”; SGA 4 XII (5.3), “special case”; SGA 4 XIV (1.11), “partial”; SGA 4 XVII (4.3.1), “special case”; SGA 4 XVII (5.1.4), “special case”; SGA 4½ Arcata (2.3.6), “more general”; SGA 4½ Th finitude (1.3), “special case”.

Let Λ be a ring and let f : X ⟶ Y be an integral morphism of schemes. Then the higher direct images of every sheaf of Λ-modules F on the small étale site of X vanish: R^q f_* F = 0 for every q > 2 · 0.

theorem isZero_rightDerived_etaleModuleDirectImage_of_isIntegralHom (Λ : Type r) [Ring Λ]
    {X Y : Scheme.{u}} (f : X ⟶ Y) [IsIntegralHom f] (F : EtaleModules.{u, r} Λ X) (q : ℕ)
    (hq : 2 * 0 < q) :
    IsZero (((etaleModuleDirectImage.{u, r} Λ f).rightDerived q).obj F)

Stacks.EtaleCohomology.ProperFibreDimension.isZero_rightDerived_etaleAbelianDirectImage_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperFibreDimensionBound.lean, line 394. The attribute's comment: “the dévissage, with the curve input still open. PROVED: the tag's own conclusion, R^q f_* F = 0 for every q > 2n and every abelian TORSION sheaf F, for f proper in the source's sense (IsProper together with QuasiCompact, since Mathlib's IsProper asks only for locally of finite type) with every fibre of dimension at most n. The content proved outright is the Leray/composition step that makes the bounds add, 2(n-1) + 2 = 2n: that is hasEtaleDirectImageBound_comp, and under it isZero_rightDerived_comp, obtained from the Grothendieck spectral sequence of Tag 015N with no hypothesis beyond Tag 015M's acyclicity criterion. The base case n = 0 is proved outright as well: Tag 02LS, proved in this library by isFinite_iff_isProper_and_forall_finite_preimage_singleton together with finite_preimage_singleton_of_topologicalKrullDim_fiber_le_zero, makes f finite and therefore integral, and Tag 03QP (2) then gives the vanishing. ASSUMED, about the given f and nothing more: hdev, for n >= 1, a factorisation f = h ≫ g in which h already has the bound 2(n-1) and the curve leg g has the bound 2 — the latter is the conclusion of Tag 03SC (via Tag 0DDF), and TAG 03SC IS OWNED BY ANOTHER ITEM OF THIS PROJECT and is deliberately NOT formalized here, while the existence of the factorisation is the geometry of the source's steps 5-7 (reduction, normalization, graph of a rational function in X x P^1), also not available in this repository. The general bound is therefore NOT proved unconditionally.”. References: SGA 4 XVII (5.1.4), “special case”; SGA 4 XVII (5.2.8.1), “special case”; SGA 4½ Cycle (2.3.1), “special case”.

[Stacks, Tag 095U], its dévissage. Let f : X ⟶ Y be a proper morphism of schemes — in the source's sense, so [IsProper f] together with [QuasiCompact f], Mathlib's IsProper asking only for locally of finite type — all of whose fibres have dimension at most n. Then for any abelian torsion sheaf F on X_étale and every q > 2n,

R^q f_* F = 0.

The case n = 0 is proved outright: [Stacks, Tag 02LS] makes f finite, by AlgebraicGeometry.isFinite_iff_isProper_and_forall_finite_preimage_singleton fed with AlgebraicGeometry.finite_preimage_singleton_of_topologicalKrullDim_fiber_le_zero, finiteness gives integrality, and the vanishing then comes from [Stacks, Tag 03QP] (2), already in this library, through hasEtaleDirectImageBound_zero_of_isIntegralHom.

One input about f itself is assumed, and nothing else:

• hdev is the outcome of the source's dévissage for f when n ≥ 1: a factorisation f = h ≫ g in which h already satisfies the bound in dimension n - 1 and g, the curve leg, satisfies the bound 2. The bound for the curve leg is the conclusion of [Stacks, Tag 03SC] (through [Stacks, Tag 0DDF]); Tag 03SC is deliberately not formalized here. The existence of the factorisation is the geometry of the source's steps 5–7 (reduction, normalization, and the graph of a rational function in X × P^1), which is likewise unavailable here, and which is why it is a hypothesis about f rather than a statement proved for every proper f.

What the file proves is the step that turns that input into the bound: the Leray/composition bound hasEtaleDirectImageBound_comp, and under it the purely homological isZero_rightDerived_comp.

theorem isZero_rightDerived_etaleAbelianDirectImage_of_isProper
    (f : X ⟶ Y) [IsProper f] [QuasiCompact f] (n : ℕ) (hdim : HasFibreDimLE f n)
    (hdev : 1 ≤ n → ∃ (C : Scheme.{u}) (h : X ⟶ C) (g : C ⟶ Y), f = h ≫ g ∧
      QuasiCompact h ∧ QuasiSeparated h ∧
      HasEtaleDirectImageBound h (2 * (n - 1)) ∧ HasEtaleDirectImageBound g 2)
    (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})
    (hF : torsionSheafProperty X.smallEtaleTopology F) (q : ℕ) (hq : 2 * n < q) :
    IsZero (((etaleAbelianDirectImage.{u, u + 1} f).rightDerived q).obj F)

Declarations registered for the tag:

• Stacks.EtaleCohomology.isZero_rightDerived_etaleAbelianDirectImage_of_isProper, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ProperDirectImageVanishingRegistration.lean, line 23 (StacksFrontier): “f proper (Mathlib's IsProper, which implies quasi-compact) with every fibre of topological Krull dimension at most n, F any abelian torsion sheaf on the small étale site of X: R^q f_* F = 0 for all q > 2n”.

Tag 0F0C, a lemma

Tag 0F0C in the Stacks Project.

Stacks.EtaleCohomology.isIso_etaleModuleDerivedBaseChangeUnbounded_app_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperBaseChange/ModuleProperBaseChangeUnbounded.lean, line 404. The attribute's comment: “for every commutative ring Λ with n = 0 in Λ for some n ≥ 1, which contains the source's Λ = ℤ/nℤ”. References: SGA 4 XVII (4.3.2), “special case”.

Stacks Project, Tag 0F0C. Let f : X ⟶ Y be a proper morphism of schemes, g : Y' ⟶ Y a morphism of schemes, and X' = Y' ×_Y X, given as a cartesian square sq with projections f' : X' ⟶ Y' and g' : X' ⟶ X. Let Λ be a commutative ring in which some integer n ≥ 1 is zero — for instance ℤ/nℤ. Then for every object E of the unbounded derived category D(X_étale, Λ) the base change map g^{-1} Rf_* E ⟶ Rf'_* (g')^{-1} E of [Stacks, Tag 0A2A] (Stacks.EtaleCohomology.etaleModuleDerivedBaseChangeUnbounded) is an isomorphism.

theorem isIso_etaleModuleDerivedBaseChangeUnbounded_app_of_isProper (Λ : Type) [CommRing Λ]
    (hΛ : ∃ n : ℕ, 0 < n ∧ (n : Λ) = 0) {X Y X' Y' : Scheme.{u}} (f : X ⟶ Y) [IsProper f]
    {g : Y' ⟶ Y} {g' : X' ⟶ X} {f' : X' ⟶ Y'} (sq : IsPullback g' f' f g)
    (E : DerivedCategory (EtaleModules Λ X)) :
    IsIso ((etaleModuleDerivedBaseChangeUnbounded Λ f g g' f' sq.w).app E)

Tag 0F0E, a lemma

Tag 0F0E in the Stacks Project.

Stacks.EtaleCohomology.DerivedTensorConstant.nonempty_derivedSections_tensor_derivedConstantSheaf_iso, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/DerivedGlobalSectionsTensorConstant.lean, line 474.

[Stacks, Tag 0F0E]. Let X be a quasi-compact and quasi-separated scheme, let E ∈ D⁺(X_étale) and K ∈ D⁺(ℤ). Then

RΓ(X, E ⊗_ℤ^L K̲) ≅ RΓ(X, E) ⊗_ℤ^L K.

Here abelian sheaves are sheaves of modules over the integers lifted to the coefficient universe, E ⊗_ℤ^L K̲ is the derived tensor product of E with the derived constant sheaf on K, RΓ(X, -) is any right derived functor of the global sections on the unbounded derived category, and the right-hand side is the derived tensor product of D(ℤ).

The isomorphism is the composite of the canonical identifications: E is represented by a bounded below complex I of injective sheaves and K by a bounded below complex P of flat modules (ℤ has global dimension one), so that E ⊗_ℤ^L K̲ = I ⊗ P̲ and RΓ(X, E) ⊗_ℤ^L K = Γ(X, I) ⊗ P; the comparison Γ(X, I) ⊗ P ⟶ Γ(X, I ⊗ P̲) is an isomorphism of complexes (totalTensorIso), and Γ(X, I ⊗ P̲) ⟶ RΓ(X, I ⊗ P̲) is an isomorphism because I ⊗ P̲ is a bounded below complex of acyclic sheaves (rightAcyclic_tensorObj_X). Both use that X is quasi-compact and quasi-separated, through the commutation of étale cohomology with filtered colimits (Stacks 03Q5): a flat module is the filtered colimit of free modules of finite rank (Stacks 058G), for which Hᵖ(X, F ⊗ M̲) = Hᵖ(X, F) ⊗ M is additivity.

theorem nonempty_derivedSections_tensor_derivedConstantSheaf_iso
    (RΓ : DerivedCategory (Sh X) ⥤ DerivedCategory (ModuleCat.{u + 1} (ULift.{u + 1} ℤ)))
    (α : (Γm X).mapHomotopyCategory (ComplexShape.up ℤ) ⋙ DerivedCategory.Qh ⟶
      DerivedCategory.Qh ⋙ RΓ)
    [RΓ.IsRightDerivedFunctor α (HomotopyCategory.quasiIso (Sh X) (ComplexShape.up ℤ))]
    (E : DerivedCategory (Sh X)) (a : ℤ) [E.IsGE a]
    (K : DerivedCategory (ModuleCat.{u + 1} (ULift.{u + 1} ℤ))) (b : ℤ) [K.IsGE b] :
    letI := CochainComplex.KFlatMonoidal.derivedMonoidal (ModuleCat.{u + 1} (ULift.{u + 1} ℤ))
      (CochainComplex.DerivedTensor.OneSided.exists_kFlat_quasiIso (ULift.{u + 1} ℤ))
    Nonempty (RΓ.obj (E ⊗ (Stacks.ModuleSheaf.derivedConstantSheafComplexes.{t, t', u + 1, u, u + 1}
        X.smallEtaleTopology).obj K) ≅ RΓ.obj E ⊗ K)

Stacks.EtaleCohomology.isIso_etaleTensorConstComparison, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/GlobalSectionsTensorConstant.lean, line 594. The attribute's comment: “steps 6-8 of the proof only: the case where E in D⁺(X_étale) is a single abelian sheaf in one degree and K in D⁺(ℤ) is a single flat ℤ-module in one degree, both carried as explicit hypotheses, so that the assertion reads H^p(X, E ⊗_ℤ K̲) = H^p(X, E) ⊗_ℤ K; the reductions of steps 1-5 and the statement for general E and K are not proved here”.

Stacks 0F0E, the case of a single sheaf and a single flat module. Let X be a quasi-compact and quasi-separated scheme, ℰ an abelian sheaf on the small étale site of X and M a flat ℤ-module. Then the canonical map

H^p(X, ℰ) ⊗_ℤ M ⟶ H^p(X, ℰ ⊗_ℤ M̲)

is an isomorphism, where M̲ is the constant sheaf on M.

This is steps 6 to 8 of the source's proof. M is written as the filtered colimit of its finitely generated subgroups, which are free of finite rank because M is torsion free; the comparison is invertible there by additivity of cohomology over a finite direct sum; and both sides commute with the filtered colimit, the left because - ⊗_ℤ M is a left adjoint and the right by [Stacks, Tag 03Q5].

theorem isIso_etaleTensorConstComparison (X : Scheme.{u})
    [CompactSpace X] [QuasiSeparatedSpace X] (p : ℕ)
    (E : Sheaf X.smallEtaleTopology AddCommGrpCat.{max (u + 1) v})
    (M : AddCommGrpCat.{max (u + 1) v}) [Module.Flat ℤ (M : Type (max (u + 1) v))] :
    IsIso (etaleTensorConstComparison X p E M)

Tag 0F0F, a lemma

Tag 0F0F in the Stacks Project.

Stacks.EtaleCohomology.isIso_etalePushforwardProjectionMap_of_isProper_of_isGE, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperProjectionFormula/BoundedBelowTorsionProjectionFormula.lean, line 41.

Projection formula for a proper morphism and a bounded below object with torsion cohomology [Stacks, Tag 0F0F]. Let f : X ⟶ Y be a proper morphism of schemes, let E ∈ D^+(X_étale) have torsion cohomology sheaves and let K ∈ D^+(Y_étale). Then the projection map Rf_*E ⊗^L_ℤ K ⟶ Rf_*(E ⊗^L_ℤ f^{-1}K) is an isomorphism in D(Y_étale, ℤ).

theorem isIso_etalePushforwardProjectionMap_of_isProper_of_isGE {X Y : Scheme.{u}} (f : X ⟶ Y)
    [IsProper f] (E : DerivedCategory (EtaleModules.{u, 0} ℤ X)) (a : ℤ) (hE : E.IsGE a)
    (hT : ∀ n, IsTorsionEtaleModule.{u, 0} ℤ X ((DerivedCategory.homologyFunctor _ n).obj E))
    (K : DerivedCategory (EtaleModules.{u, 0} ℤ Y)) (b : ℤ) (hK : K.IsGE b) :
    IsIso (etalePushforwardProjectionMap.{u, 0} ℤ f E K)

Sources

• Section 59.92, Tag 0A5I
• AxiomMath/Stacks at 5ed82d14

LeanStacks.EtaleCohomology.bijective_etaleCohomologyRestrictionMap_scalarExtensionFst_of_isProper · Stacks.EtaleCohomology.isZero_rightDerived_etaleModuleDirectImage_of_isIntegralHom · Stacks.EtaleCohomology.ProperFibreDimension.isZero_rightDerived_etaleAbelianDirectImage_of_isProper · Stacks.EtaleCohomology.isZero_rightDerived_etaleAbelianDirectImage_of_isProper · Stacks.EtaleCohomology.isIso_etaleModuleDerivedBaseChangeUnbounded_app_of_isProper · Stacks.EtaleCohomology.DerivedTensorConstant.nonempty_derivedSections_tensor_derivedConstantSheaf_iso · Stacks.EtaleCohomology.isIso_etaleTensorConstComparison · Stacks.EtaleCohomology.isIso_etalePushforwardProjectionMap_of_isProper_of_isGE

Section59.93 Local acyclicity#

Section 59.93 of the Stacks Project, Tag 0GJM. 5 tags here are carried by 17 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0GJN, an equation

Tag 0GJN in the Stacks Project.

AlgebraicGeometry.vanishingCyclesMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/SpecializationMap.lean, line 461. The attribute's comment: “for an arbitrary scheme over the strict henselization”.

Stacks 0GJN. The canonical map α_{K,x,t} : K_x ⟶ RΓ(T, K) from the stalk at a geometric point x of X of an object K of the unbounded derived category of abelian sheaves on the small étale site of X, to the unbounded derived global sections of K on a scheme T over the strict henselization Spec 𝒪^{sh}_{X,x}, natural in K.

For a morphism of schemes f : X ⟶ S and a geometric point t of Spec 𝒪^{sh}_{S,f(x)} the variety of vanishing cycles of f at x is the fibre product F_{x,t} = t ×_{Spec 𝒪^{sh}_{S,f(x)}} Spec 𝒪^{sh}_{X,x} over Spec 𝒪^{sh}_{X,x}, and the displayed map of the source is this one for that T and the second projection; the construction uses nothing about T beyond its morphism to the strict henselization.

Since the strict henselization is the initial étale neighbourhood of x, pulling back germs along T ⟶ Spec 𝒪^{sh}_{X,x} gives the map etaleStalkToSectionsAb on abelian sheaves. The stalk functor is exact, so its action on unbounded derived categories is a right derived functor of itself, and the map on sheaves extends uniquely to one of derived categories.

def vanishingCyclesMap :
    smallEtaleDerivedStalkAb x.hom ⟶
      smallEtaleDerivedPullbackAb
          (q ≫ strictHenselizationToBase B X hX x) ⋙
        smallEtaleDerivedGlobalSections T

AlgebraicGeometry.vanishingCyclesMap_fac_app, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/SpecializationMap.lean, line 481. The attribute's comment: “the derived map factors the map on abelian sheaves”.

Stacks 0GJN. On every complex of abelian sheaves on the small étale site of X, the canonical map to the derived global sections over T factors the degreewise map on sheaves through the unit of the derivation of global sections, read through the identifications of the degreewise actions of the stalk and of inverse image with their actions on derived categories.

Since the degreewise action of the stalk functor is a right derived functor of itself, this property determines the map.

theorem vanishingCyclesMap_fac_app
    (J : HomotopyCategory (Sheaf (smallEtaleTopology X) Ab.{u + 1}) (.up ℤ)) :
    (smallEtaleStalkAb x.hom).mapDerivedCategoryFactorsh.inv.app J ≫
        (vanishingCyclesMap B X hX x q).app (DerivedCategory.Qh.obj J) =
      DerivedCategory.Qh.map ((Stacks.DerivedFunctor.mapHomotopyCategoryOfComp
          (smallEtaleStalkAb x.hom)
          (smallEtalePullbackAb
            (q ≫ strictHenselizationToBase B X hX x))
          (Sheaf.Γ (smallEtaleTopology T) Ab.{u + 1})
          (etaleStalkToSectionsAb B X hX x q)).app J) ≫
        (smallEtaleDerivedGlobalSectionsUnit T).app
            (((smallEtalePullbackAb
              (q ≫ strictHenselizationToBase B X hX x)).mapHomotopyCategory (.up ℤ)).obj J) ≫
          (smallEtaleDerivedGlobalSections T).map
            ((smallEtalePullbackAb
              (q ≫ strictHenselizationToBase B X hX x)).mapDerivedCategoryFactorsh.inv.app J)

AlgebraicGeometry.vanishingCyclesFibreMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/SpecializationMap.lean, line 537.

Stacks 0GJN. The canonical map

α_{K,x,t} : K_x ⟶ RΓ(F_{x,t}, K)

from the stalk at a geometric point x of X of an object K of the unbounded derived category of abelian sheaves on the small étale site of X, to the unbounded derived sections of K on the variety of vanishing cycles F_{x,t} of a morphism f : X ⟶ S at x and a geometric point t of Spec 𝒪^{sh}_{S,f(x)}, natural in K.

The stalk of K at x is the derived sections of K over the strictly henselian Spec 𝒪^{sh}_{X,x}, and the map is pullback of cohomology along the projection F_{x,t} ⟶ Spec 𝒪^{sh}_{X,x}.

def vanishingCyclesFibreMap :
    smallEtaleDerivedStalkAb x.hom ⟶
      smallEtaleDerivedPullbackAb (vanishingCyclesFibreToSource B hX hS f x t) ⋙
        smallEtaleDerivedGlobalSections (vanishingCyclesFibre B hX hS f x t)

AlgebraicGeometry.smallEtaleDerivedStalkToSections, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StalkToSectionsRestriction.lean, line 98. The attribute's comment: “for an arbitrary scheme over the strict henselization”.

Stacks 0GJN. The canonical map α_{K,x} : K_x ⟶ RΓ(T, K) from the stalk at a geometric point x of X of an object K of the unbounded derived category D(X_étale) of abelian sheaves on the small étale site of X, to the unbounded derived global sections RΓ(T, K) = RΓ(T_étale, g⁻¹K) of K on a scheme T over the strict henselization Spec 𝒪^{sh}_{X,x}, with g : T ⟶ Spec 𝒪^{sh}_{X,x} ⟶ X the structure morphism, natural in K.

The derived global sections on T are the total right derived functor of global sections on the whole unbounded homotopy category, which exists by Stacks.EtaleCohomology.smallEtaleGlobalSectionsAb_hasRightDerivedFunctor. Since the strict henselization is the initial étale neighbourhood of x, pulling back germs along T ⟶ Spec 𝒪^{sh}_{X,x} gives the map etaleStalkToSectionsAb on abelian sheaves; the stalk functor is exact, so the map on sheaves extends uniquely to one of derived categories. AlgebraicGeometry.smallEtaleDerivedStalkToSections_eq_restriction identifies it with the identification K_x = RΓ(Spec 𝒪^{sh}_{X,x}, K) followed by pulling back cohomology along T ⟶ Spec 𝒪^{sh}_{X,x}.

def smallEtaleDerivedStalkToSections :
    smallEtaleDerivedStalkAb x.hom ⟶
      smallEtaleDerivedPullbackAb (q ≫ strictHenselizationToBase B X hX x) ⋙
        smallEtaleDerivedGlobalSections T

AlgebraicGeometry.smallEtaleDerivedStalkToSections_fac_app, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StalkToSectionsRestriction.lean, line 112. The attribute's comment: “the derived map factors the map on abelian sheaves”.

Stacks 0GJN. On every complex of abelian sheaves on the small étale site of X, the canonical map to the derived global sections over T factors the degreewise map on sheaves through the unit of the derivation of global sections, read through the identifications of the degreewise actions of the stalk and of inverse image with their actions on derived categories.

Since the degreewise action of the stalk functor is a right derived functor of itself, this property determines the map.

theorem smallEtaleDerivedStalkToSections_fac_app
    (J : HomotopyCategory (Sheaf (smallEtaleTopology X) Ab.{u + 1}) (.up ℤ)) :
    (smallEtaleStalkAb x.hom).mapDerivedCategoryFactorsh.inv.app J ≫
        (smallEtaleDerivedStalkToSections B X hX x q).app (DerivedCategory.Qh.obj J) =
      DerivedCategory.Qh.map ((Stacks.DerivedFunctor.mapHomotopyCategoryOfComp
          (smallEtaleStalkAb x.hom)
          (smallEtalePullbackAb
            (q ≫ strictHenselizationToBase B X hX x))
          (Sheaf.Γ (smallEtaleTopology T) Ab.{u + 1})
          (etaleStalkToSectionsAb B X hX x q)).app J) ≫
        (smallEtaleDerivedGlobalSectionsUnit T).app
            (((smallEtalePullbackAb
              (q ≫ strictHenselizationToBase B X hX x)).mapHomotopyCategory (.up ℤ)).obj J) ≫
          (smallEtaleDerivedGlobalSections T).map
            ((smallEtalePullbackAb
              (q ≫ strictHenselizationToBase B X hX x)).mapDerivedCategoryFactorsh.inv.app J)

AlgebraicGeometry.stalkToVanishingCyclesFibreSections, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StalkToSectionsRestriction.lean, line 156.

Stacks 0GJN. The canonical map

α_{K,x,t} : K_x ⟶ RΓ(F_{x,t}, K)

from the stalk at a geometric point x of X of an object K of the unbounded derived category D(X_étale) of abelian sheaves on the small étale site of X, to the unbounded derived sections RΓ(F_{x,t}, K) = RΓ((F_{x,t})_étale, g⁻¹K) of K on the variety of vanishing cycles F_{x,t} = t ×_{Spec 𝒪^{sh}_{S,f(x)}} Spec 𝒪^{sh}_{X,x} of a morphism f : X ⟶ S at x and a geometric point t of Spec 𝒪^{sh}_{S,f(x)}, with g : F_{x,t} ⟶ X the canonical morphism, natural in K.

The derived global sections on F_{x,t} exist by Stacks.EtaleCohomology.smallEtaleGlobalSectionsAb_hasRightDerivedFunctor. The stalk of K at x is the derived sections of K over the strictly henselian Spec 𝒪^{sh}_{X,x} (AlgebraicGeometry.smallEtaleDerivedStalkIsoStrictHenselizationSections), and the map is pullback of cohomology along the projection F_{x,t} ⟶ Spec 𝒪^{sh}_{X,x} (AlgebraicGeometry.stalkToVanishingCyclesFibreSections_eq_restriction).

def stalkToVanishingCyclesFibreSections :
    smallEtaleDerivedStalkAb x.hom ⟶
      smallEtaleDerivedPullbackAb (vanishingCyclesFibreToSource B hX hS f x t) ⋙
        smallEtaleDerivedGlobalSections (vanishingCyclesFibre B hX hS f x t)

AlgebraicGeometry.smallEtaleDerivedStalkIsoStrictHenselizationSections, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StalkToSectionsRestriction.lean, line 183. The attribute's comment: “K_x = RΓ(Spec 𝒪^{sh}_{X,x}, K), since 𝒪^{sh}_{X,x} is strictly henselian”.

The stalk is the derived sections over the strict henselization. For every object K of the unbounded derived category of abelian sheaves on the small étale site of X, the canonical map K_x ⟶ RΓ(Spec 𝒪^{sh}_{X,x}, p⁻¹K) is an isomorphism, natural in K: on sheaves it is F_x = Γ(Spec 𝒪^{sh}_{X,x}, p⁻¹F), and global sections over the strictly henselian Spec 𝒪^{sh}_{X,x} are exact, so no boundedness of K is needed.

def smallEtaleDerivedStalkIsoStrictHenselizationSections :
    smallEtaleDerivedStalkAb x.hom ≅
      smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x) ⋙
        smallEtaleDerivedGlobalSections (strictHenselization B X hX x)

AlgebraicGeometry.stalkToVanishingCyclesFibreSections_eq_restriction, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StalkToSectionsRestriction.lean, line 284. The attribute's comment: “the map is obtained by pulling back cohomology along F_{x,t} ⟶ Spec 𝒪^{sh}_{X,x}, using K_x = RΓ(Spec 𝒪^{sh}_{X,x}, K)”.

Stacks 0GJN: the map α_{K,x,t} is pullback of cohomology along F_{x,t} ⟶ Spec 𝒪^{sh}_{X,x}. The canonical map α_{K,x,t} : K_x ⟶ RΓ(F_{x,t}, K) at the variety of vanishing cycles F_{x,t} = t ×_{Spec 𝒪^{sh}_{S,f(x)}} Spec 𝒪^{sh}_{X,x} is the identification K_x = RΓ(Spec 𝒪^{sh}_{X,x}, K) followed by restriction of unbounded derived global sections along the second projection F_{x,t} ⟶ Spec 𝒪^{sh}_{X,x}, read through the composition isomorphism of inverse images.

theorem stalkToVanishingCyclesFibreSections_eq_restriction :
    stalkToVanishingCyclesFibreSections B hX hS f x t =
      (smallEtaleDerivedStalkIsoStrictHenselizationSections B X hX x).hom ≫
        Functor.whiskerLeft (smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x))
          (smallEtaleDerivedGlobalSectionsRestriction
            (vanishingCyclesFibreToStrictHenselization B hX hS f x t)) ≫
        (Functor.associator _ _ _).inv ≫
          Functor.whiskerRight
            (Functor.mapDerivedCategoryCompIsoOfCompIso
              (smallEtalePullbackAbCompIso (vanishingCyclesFibreToStrictHenselization B hX hS f x t)
                (strictHenselizationToBase B X hX x))).hom
            (smallEtaleDerivedGlobalSections (vanishingCyclesFibre B hX hS f x t))

Tag 0GJP, a definition

Tag 0GJP in the Stacks Project.

AlgebraicGeometry.localAcyclicity_iff_isIso_vanishingCyclesFibreMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/Basic.lean, line 221.

Let f : X ⟶ S be a morphism of schemes and let K be an object of D(X_étale). The five notions of the definition are, in terms of the canonical maps α_{K,x,t} : K_x ⟶ RΓ(F_{x,t}, K) of [Stacks, Tag 0GJN]:

1. f is locally acyclic at a geometric point x of X relative to K when α_{K,x,t} is an isomorphism for every geometric point t of Spec 𝒪^{sh}_{S,f(x)};
2. f is locally acyclic relative to K when that holds at every geometric point x of X;
3. f is universally locally acyclic relative to K when for every morphism g : S' ⟶ S of schemes the base change X ×_S S' ⟶ S' is locally acyclic relative to the inverse image of K;
4. f is locally acyclic when for every geometric point x of X and every positive integer n prime to the characteristic of κ(x) it is locally acyclic at x relative to the constant sheaf ℤ/nℤ;
5. f is universally locally acyclic when every base change of f is locally acyclic.

The geometric points t are not assumed to be algebraic geometric points of Spec 𝒪^{sh}_{S,f(x)}: GeometricPoint asks only for a morphism from the spectrum of an algebraically closed field.

The varieties of vanishing cycles are formed inside a big étale site, so clauses (1), (2) and (4) carry one; clauses (3) and (5) quantify over all morphisms g : S' ⟶ S and over all big étale sites containing X ×_S S' and S', of which there is always at least one (AlgebraicGeometry.exists_bigEtaleSite_contains_pullback).

public theorem localAcyclicity_iff_isIso_vanishingCyclesFibreMap (B : BigEtaleSite.{u})
    {X S : Scheme.{u}} (hX : B.Contains X) (hS : B.Contains S) (f : X ⟶ S)
    (K : smallEtaleDerivedCategory X) :
    (∀ x : GeometricPoint X, IsLocallyAcyclicAt B hX hS f K x ↔
        ∀ t : GeometricPoint (strictHenselization B S hS (x.map f)),
          IsIso ((vanishingCyclesFibreMap B hX hS f x t).app K)) ∧
      (IsLocallyAcyclicRelative B hX hS f K ↔
        ∀ (x : GeometricPoint X)
          (t : GeometricPoint (strictHenselization B S hS (x.map f))),
          IsIso ((vanishingCyclesFibreMap B hX hS f x t).app K)) ∧
      (IsUniversallyLocallyAcyclicRelative f K ↔
        ∀ (S' : Scheme.{u}) (g : S' ⟶ S) (B' : BigEtaleSite.{u})
          (hX' : B'.Contains (pullback f g)) (hS' : B'.Contains S')
          (x' : GeometricPoint (pullback f g))
          (t' : GeometricPoint
            (strictHenselization B' S' hS' (x'.map (pullback.snd f g)))),
          IsIso ((vanishingCyclesFibreMap B' hX' hS'
              (pullback.snd f g) x' t').app
            ((smallEtaleDerivedPullbackAb (pullback.fst f g)).obj K))) ∧
      (IsLocallyAcyclic B hX hS f ↔
        ∀ (x : GeometricPoint X) (n : ℕ), 0 < n →
          (ringChar x.residueField = 0 ∨ n.Coprime (ringChar x.residueField)) →
            ∀ t : GeometricPoint (strictHenselization B S hS (x.map f)),
              IsIso ((vanishingCyclesFibreMap B hX hS f x t).app
                (smallEtaleConstantZModComplex X n))) ∧
      (IsUniversallyLocallyAcyclic f ↔
        ∀ (S' : Scheme.{u}) (g : S' ⟶ S) (B' : BigEtaleSite.{u})
          (hX' : B'.Contains (pullback f g)) (hS' : B'.Contains S')
          (x' : GeometricPoint (pullback f g)) (n : ℕ), 0 < n →
          (ringChar x'.residueField = 0 ∨ n.Coprime (ringChar x'.residueField)) →
            ∀ t' : GeometricPoint
              (strictHenselization B' S' hS' (x'.map (pullback.snd f g))),
              IsIso ((vanishingCyclesFibreMap B' hX' hS'
                  (pullback.snd f g) x' t').app
                (smallEtaleConstantZModComplex (pullback f g) n)))

Tag 0GJQ, a proposition

Tag 0GJQ in the Stacks Project.

Stacks.EtaleCohomology.SmoothBaseChange.isUniversallyLocallyAcyclic_of_smooth_unconditional, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/SmoothBaseChange/LocalAcyclicity/SmoothMorphismsRegistered.lean, line 38. References: SGA 4½ Arcata (5.2.1), “more general”; SGA 7 I (2.4), “partial”.

Smooth morphisms are universally locally acyclic ([Stacks, Proposition 0GJQ]). Let f : X ⟶ S be a smooth morphism of schemes. Then f is universally locally acyclic: for every morphism S' ⟶ S and every big étale site containing X ×_S S' and S', the base change X ×_S S' ⟶ S' of f is locally acyclic there.

theorem isUniversallyLocallyAcyclic_of_smooth_unconditional {X S : Scheme.{u}} (f : X ⟶ S)
    [Smooth f] : IsUniversallyLocallyAcyclic f

Tag 0GJR, a lemma

Tag 0GJR in the Stacks Project.

AlgebraicGeometry.isLocallyAcyclicRelative_of_isLocallyConstant, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/LocallyConstantCoefficients.lean, line 382. The attribute's comment: “the dévissage from constant ℤ/nℤ coefficients to a locally constant prime-to-residue-characteristic torsion sheaf: the stalk is the filtered colimit of its finite subgroups and each of those is a finite direct sum of cyclic groups. The two strict-henselization inputs are hypotheses — hdep, that the canonical map depends only on the inverse image on Spec O^sh_{X,x}, and hconst, that a locally constant sheaf is constant there with value its stalk — as are hcolim, which is Tag 03Q5 for the affine variety of vanishing cycles, and hadd, additivity of RGamma”.

Stacks 0GJR. Let f : X ⟶ S be a morphism of schemes and let F be a locally constant abelian sheaf on X_étale whose stalk at every geometric point x is torsion with every element order prime to the characteristic of the residue field at x. If f is locally acyclic, then f is locally acyclic relative to F.

What is proved here is the dévissage: the reduction of the locally constant torsion coefficient sheaf F to the constant sheaves ℤ/nℤ that local acyclicity speaks about. The stalk is written as the filtered colimit of its finite subgroups, each of those splits as a finite direct sum of cyclic groups of prime-to-p order, and local acyclicity is applied to each summand.

Four hypotheses are supplied rather than derived.

• hdep and hconst are the two facts about the strict henselization that the source's proof rests on, each stated in its own right. hdep is the consequence of K_x = RΓ(Spec 𝒪^{sh}_{X,x}, K) that the canonical map only sees the inverse image of K on Spec 𝒪^{sh}_{X,x}; hconst is that a locally constant sheaf has constant inverse image there, with value its stalk.
• hcolim is Tag 03Q5 for the affine variety of vanishing cycles, in the form the argument consumes it: the iso-locus of the canonical map on constant coefficients is closed under filtered colimits of the coefficient group.
• hadd is additivity of unbounded derived global sections, which is formal but not an available instance.

See the module docstring for why hcolim cannot be stated as preservation of filtered colimits.

theorem isLocallyAcyclicRelative_of_isLocallyConstant
    (B : BigEtaleSite.{u}) {X S : Scheme.{u}} (hX : B.Contains X) (hS : B.Contains S)
    (f : X ⟶ S)
    (hadd : ∀ (x : GeometricPoint X)
        (t : GeometricPoint (strictHenselization B S hS (x.map f))),
      (smallEtaleDerivedGlobalSections (vanishingCyclesFibre B hX hS f x t)).Additive)
    (hcolim : ∀ (x : GeometricPoint X)
        (t : GeometricPoint (strictHenselization B S hS (x.map f))),
      ∀ {I : Type (u + 1)} [SmallCategory I] [IsFiltered I] (N : I ⥤ Ab.{u + 1})
        (c : Cocone N), IsColimit c →
        (∀ i, IsIso ((vanishingCyclesFibreMap B hX hS f x t).app
            (smallEtaleConstantComplexAb X (N.obj i)))) →
          IsIso ((vanishingCyclesFibreMap B hX hS f x t).app
            (smallEtaleConstantComplexAb X c.pt)))
    (hdep : ∀ (x : GeometricPoint X)
        (t : GeometricPoint (strictHenselization B S hS (x.map f)))
        (K K' : smallEtaleDerivedCategory X),
      ((smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x)).obj K ≅
          (smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x)).obj K') →
      (IsIso ((vanishingCyclesFibreMap B hX hS f x t).app K) ↔
        IsIso ((vanishingCyclesFibreMap B hX hS f x t).app K')))
    (hconst : ∀ (G : Sheaf X.smallEtaleTopology Ab.{u + 1}),
        Stacks.ConstantSheaf.IsLocallyConstant X.smallEtaleTopology G →
        ∀ x : GeometricPoint X,
      (smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x)).obj
          ((DerivedCategory.singleFunctor (Sheaf X.smallEtaleTopology Ab.{u + 1}) 0).obj G) ≅
        (smallEtaleDerivedPullbackAb (strictHenselizationToBase B X hX x)).obj
          (smallEtaleConstantComplexAb X ((smallEtaleStalkAb x.hom).obj G)))
    (F : Sheaf X.smallEtaleTopology Ab.{u + 1})
    (hFlc : Stacks.ConstantSheaf.IsLocallyConstant X.smallEtaleTopology F)
    (hFtors : ∀ (x : GeometricPoint X) (a : (smallEtaleStalkAb x.hom).obj F),
      ∃ n : ℕ, 0 < n ∧
        (ringChar x.residueField = 0 ∨ Nat.Coprime n (ringChar x.residueField)) ∧ n • a = 0)
    (hf : IsLocallyAcyclic B hX hS f) :
    IsLocallyAcyclicRelative B hX hS f
      ((DerivedCategory.singleFunctor (Sheaf X.smallEtaleTopology Ab.{u + 1}) 0).obj F)

AlgebraicGeometry.isLocallyAcyclicRelative_of_isLocallyConstant_unconditional, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/LocallyConstantCoefficientsUnconditional.lean, line 401.

Stacks 0GJR. Let f : X ⟶ S be a morphism of schemes and let F be a locally constant abelian sheaf on X_étale whose stalk at every geometric point x is torsion with every element order prime to the characteristic of the residue field at x. If f is locally acyclic, then f is locally acyclic relative to F.

What is proved here is the dévissage: the reduction of the locally constant torsion coefficient sheaf F to the constant sheaves ℤ/nℤ that local acyclicity speaks about. The stalk is written as the filtered colimit of its finite subgroups, each of those splits as a finite direct sum of cyclic groups of prime-to-p order, and local acyclicity is applied to each summand.

No hypothesis is supplied rather than derived. The two facts about the strict henselization that the source's proof rests on are both proved: that the canonical map only sees the inverse image of K on Spec 𝒪^{sh}_{X,x} is AlgebraicGeometry.isIso_vanishingCyclesFibreMap_app_iff_of_derivedPullbackIso, and that a locally constant sheaf has constant inverse image there, with value its stalk, is AlgebraicGeometry.strictHenselizationDerivedPullbackConstantComplexIso. Nor is the filtered colimit step a hypothesis: it is AlgebraicGeometry.isIso_vanishingCyclesFibreMap_app_constantComplex_of_isColimit, Tag 03Q5 for the variety of vanishing cycles, and neither is additivity of RΓ there, which is Stacks.EtaleCohomology.smallEtaleDerivedGlobalSections_additive.

AlgebraicGeometry.isLocallyAcyclicRelative_of_isLocallyConstant is the same statement with those four facts taken as hypotheses.

theorem isLocallyAcyclicRelative_of_isLocallyConstant_unconditional
    (B : BigEtaleSite.{u}) {X S : Scheme.{u}} (hX : B.Contains X) (hS : B.Contains S)
    (f : X ⟶ S)
    (F : Sheaf X.smallEtaleTopology Ab.{u + 1})
    (hFlc : Stacks.ConstantSheaf.IsLocallyConstant X.smallEtaleTopology F)
    (hFtors : ∀ (x : GeometricPoint X) (a : (smallEtaleStalkAb x.hom).obj F),
      ∃ n : ℕ, 0 < n ∧
        (ringChar x.residueField = 0 ∨ Nat.Coprime n (ringChar x.residueField)) ∧ n • a = 0)
    (hf : IsLocallyAcyclic B hX hS f) :
    IsLocallyAcyclicRelative B hX hS f
      ((DerivedCategory.singleFunctor (Sheaf X.smallEtaleTopology Ab.{u + 1}) 0).obj F)

Tag 0GJS, a lemma

Tag 0GJS in the Stacks Project.

AlgebraicGeometry.isLocallyAcyclicAt_of_isIntegralHom, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/BaseChangeTransfer.lean, line 101. The attribute's comment: “integral base change”.

For a cartesian square of schemes with g integral, an object K of D(X_étale) and a geometric point x' of the fibre product with image x in X, local acyclicity of f at x relative to K implies local acyclicity of f' at x' relative to (g')⁻¹K.

public theorem isLocallyAcyclicAt_of_isIntegralHom [IsIntegralHom g]
    (h : IsLocallyAcyclicAt B hX hS f K (x'.map g')) :
    IsLocallyAcyclicAt B hX' hS' f' ((smallEtaleDerivedPullbackAb g').obj K) x'

AlgebraicGeometry.isLocallyAcyclicAt_of_locallyQuasiFinite, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/BaseChangeTransfer.lean, line 113. The attribute's comment: “locally quasi-finite base change”. References: SGA 4½ Arcata (5.1.4), “more general”.

For a cartesian square of schemes with g locally of finite type and locally quasi-finite, an object K of D(X_étale) and a geometric point x' of the fibre product with image x in X, local acyclicity of f at x relative to K implies local acyclicity of f' at x' relative to (g')⁻¹K.

public theorem isLocallyAcyclicAt_of_locallyQuasiFinite [LocallyOfFiniteType g]
    [LocallyQuasiFinite g] (h : IsLocallyAcyclicAt B hX hS f K (x'.map g')) :
    IsLocallyAcyclicAt B hX' hS' f' ((smallEtaleDerivedPullbackAb g').obj K) x'

AlgebraicGeometry.isLocallyAcyclicAt_of_limit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/BaseChangeTransfer.lean, line 146. The attribute's comment: “directed inverse limit base change”. References: SGA 4½ Arcata (5.1.4), “more general”.

Let { Dᵢ } be an inverse system of schemes indexed by a cofiltered category, with affine transition morphisms, whose stages are locally of finite type and locally quasi-finite over S compatibly with the transition morphisms, let S' = lim Dᵢ, and let

X' --g'--> X
|f'        |f
S' --g---> S

be a cartesian square whose g is induced by the morphisms from the stages. For an object K of D(X_étale) and a geometric point x' of X' with image x in X, local acyclicity of f at x relative to K implies local acyclicity of f' at x' relative to (g')⁻¹K.

public theorem isLocallyAcyclicAt_of_limit (p : ∀ i : I, D.obj i ⟶ S)
    (hp : ∀ i : I, c.π.app i ≫ p i = g) (hcompat : ∀ {i j : I} (a : i ⟶ j), D.map a ≫ p j = p i)
    [∀ i : I, LocallyOfFiniteType (p i)] [∀ i : I, LocallyQuasiFinite (p i)]
    (h : IsLocallyAcyclicAt B hX hS f K (x'.map g')) :
    IsLocallyAcyclicAt B hX' hS' f' ((smallEtaleDerivedPullbackAb g').obj K) x'

AlgebraicGeometry.isLocallyAcyclicAt_strictHenselizationBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StrictHenselizationBaseChangeSite.lean, line 87. The attribute's comment: “base change along the strict henselization at a geometric point”. References: SGA 4½ Arcata (5.1.4), “special case”; SGA 4½ Arcata (5.1.6), “more general”.

Local acyclicity survives base change to the strict henselization. Let f : X ⟶ S be a morphism of schemes of a big étale site B, let K be an object of D(X_étale), let s be a geometric point of S and let x be a geometric point of the base change Y = X ×_S Spec 𝒪^{sh}_{S,s} with image y in X. If f is locally acyclic at y relative to K, then the projection Y ⟶ Spec 𝒪^{sh}_{S,s} is locally acyclic at x relative to K|_Y.

theorem isLocallyAcyclicAt_strictHenselizationBaseChange
    (x : GeometricPoint (strictHenselizationBaseChange B hS f s))
    (h : IsLocallyAcyclicAt B hX hS f K
      (x.map (strictHenselizationBaseChangeToSource B hS f s))) :
    IsLocallyAcyclicAt B (contains_strictHenselizationBaseChange B hS hX f s)
      (contains_strictHenselization_of_contains B hS s)
      (strictHenselizationBaseChangeToStrictHenselization B hS f s)
      ((smallEtaleDerivedPullbackAb
        (strictHenselizationBaseChangeToSource B hS f s)).obj K) x

AlgebraicGeometry.isLocallyAcyclicRelative_strictHenselizationBaseChange, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/StrictHenselizationBaseChangeSite.lean, line 111. The attribute's comment: “base change along the strict henselization at a geometric point”. References: SGA 4½ Arcata (5.1.6), “more general”.

Local acyclicity survives base change to the strict henselization. If f : X ⟶ S is locally acyclic relative to an object K of D(X_étale), then the projection X ×_S Spec 𝒪^{sh}_{S,s} ⟶ Spec 𝒪^{sh}_{S,s} is locally acyclic relative to the restriction of K to the base change.

theorem isLocallyAcyclicRelative_strictHenselizationBaseChange
    (hla : IsLocallyAcyclicRelative B hX hS f K) :
    IsLocallyAcyclicRelative B (contains_strictHenselizationBaseChange B hS hX f s)
      (contains_strictHenselization_of_contains B hS s)
      (strictHenselizationBaseChangeToStrictHenselization B hS f s)
      ((smallEtaleDerivedPullbackAb
        (strictHenselizationBaseChangeToSource B hS f s)).obj K)

Sources

• Section 59.93, Tag 0GJM
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.vanishingCyclesMap · AlgebraicGeometry.vanishingCyclesMap_fac_app · AlgebraicGeometry.vanishingCyclesFibreMap · AlgebraicGeometry.smallEtaleDerivedStalkToSections · AlgebraicGeometry.smallEtaleDerivedStalkToSections_fac_app · AlgebraicGeometry.stalkToVanishingCyclesFibreSections · AlgebraicGeometry.smallEtaleDerivedStalkIsoStrictHenselizationSections · AlgebraicGeometry.stalkToVanishingCyclesFibreSections_eq_restriction

Section59.94 The cospecialization map#

Section 59.94 of the Stacks Project, Tag 0GJT. 4 tags here are carried by 5 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0GJU, a lemma

Tag 0GJU in the Stacks Project.

AlgebraicGeometry.isIso_pullback_cospecializationUnit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/CospecializationMap.lean, line 255. The attribute's comment: “modulo derived-stalk conservativity and the stalk computation”.

Assume that isomorphy in unbounded derived categories of abelian sheaves on small étale sites is detected on stalks (hcons) and that the stalks of i⁻¹β_{K,s,t} are canonical maps of Tag 0GJN (hcomp). For a morphism of schemes f : X ⟶ S which is locally acyclic relative to an object K of D(X_étale), the inverse image on the closed geometric fibre X_s of the comparison β_{K,s,t} : K|_Y ⟶ Rh_*(K|_{X_t}) is an isomorphism.

theorem isIso_pullback_cospecializationUnit
    (hla : IsLocallyAcyclicRelative B hX hS f K)
    (hcons : HasDerivedStalkConservativity.{u})
    (hcomp : HasCospecializationStalkComparison B) :
    IsIso ((smallEtaleDerivedPullbackAb
      (geometricFibreToStrictHenselizationBaseChange B hS f s)).map
        (cospecializationUnit B hS f s t K))

AlgebraicGeometry.isIso_pullback_cospecializationUnit_of_isGE, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/ProperLocallyAcyclicSpecialization.lean, line 76.

[Stacks, Tag 0GJU]. Let f : X ⟶ S be a morphism of schemes, s a geometric point of S, t a geometric point of Spec 𝒪^{sh}_{S,s}, and K an object of D(X_étale) which is bounded below and relative to which f is locally acyclic. Write Y = X ×_S Spec 𝒪^{sh}_{S,s}, h : X_t ⟶ Y for the fibre over t and i : X_s ⟶ Y for the closed geometric fibre. Then the inverse image i⁻¹β_{K,s,t} of the comparison β_{K,s,t} : K|_Y ⟶ Rh_*(K|_{X_t}) is an isomorphism.

theorem isIso_pullback_cospecializationUnit_of_isGE (K : smallEtaleDerivedCategory X) (m : ℤ)
    (hK : K.IsGE m) (hla : IsLocallyAcyclicRelative B hX hS f K) :
    IsIso ((smallEtaleDerivedPullbackAb
      (geometricFibreToStrictHenselizationBaseChange B hS f s)).map
        (cospecializationUnit B hS f s t K))

Tag 0GJV, a lemma

Tag 0GJV in the Stacks Project.

AlgebraicGeometry.specializationStalk_comp_baseChange_comp_cospecializationMap_of_isGE, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/CospecializationSquareLocallyAcyclic.lean, line 92.

[Stacks, Tag 0GJV]. Let f : X ⟶ S be a quasi-compact and quasi-separated morphism of schemes, s a geometric point of S, t a geometric point of Spec 𝒪^{sh}_{S,s}, and K an object of D(X_étale) which is bounded below and relative to which f is locally acyclic, so that the cospecialization map cosp : RΓ(X_t, K) ⟶ RΓ(X_s, K) is defined (Stacks 0GJU, AlgebraicGeometry.isIso_pullback_cospecializationUnit_of_isGE). Then the square

(Rf_*K)_s ----------> RΓ(X_s, K)
   | sp                    ^ cosp
   v                       |
(Rf_*K)_t ----------> RΓ(X_t, K)

commutes: sp followed by the base change map of Stacks 0GIX at t and by cosp is the base change map of Stacks 0GIX at s. Here X_s and X_t are the geometric fibres of f at s and at the image of t in S, and the stalk of Rf_*K at t is taken at that image.

theorem specializationStalk_comp_baseChange_comp_cospecializationMap_of_isGE
    [QuasiCompact f] [QuasiSeparated f] (K : smallEtaleDerivedCategory X) (m : ℤ)
    (hK : K.IsGE m) (hla : IsLocallyAcyclicRelative B hX hS f K) :
    haveI := isIso_pullback_cospecializationUnit_of_isGE B hX hS f s t K m hK hla
    (smallEtaleDerivedSpecializationStalkAbAt B S hS s t).app
          ((smallEtaleDerivedPushforwardAb f).obj K) ≫
        (derivedPushforwardStalkToStrictHenselizationSections B hS f
          (t.map (strictHenselizationToBase B S hS s))).app K ≫
        (strictHenselizationSectionsToFibreSections B hS f
          (t.map (strictHenselizationToBase B S hS s))).app K ≫
        cospecializationMapGeometricFibre B hS f s t K =
      (derivedPushforwardStalkToStrictHenselizationSections B hS f s).app K ≫
        (strictHenselizationSectionsToFibreSections B hS f s).app K

Tag 0GJW, a lemma

Tag 0GJW in the Stacks Project.

AlgebraicGeometry.isIso_specializationStalk_and_cospecializationMap_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/ProperLocallyAcyclicSpecialization.lean, line 127. References: SGA 4 XVI (2.3), “special case”; SGA 4½ Arcata (5.3.1), “more general”; SGA 4½ Th finitude (5.2.4), “for proper f”.

[Stacks, Tag 0GJW]. Let f : X ⟶ S be a morphism of schemes and K an object of D(X_étale) such that

1. K is bounded below,
2. f is locally acyclic relative to K,
3. f is proper, and
4. K has torsion cohomology sheaves.

Then for every geometric point s of S and every geometric point t of Spec 𝒪^{sh}_{S,s} the specialization map sp : (Rf_*K)_s ⟶ (Rf_*K)_t is an isomorphism, and the cospecialization map cosp : RΓ(X_t, K) ⟶ RΓ(X_s, K) — defined because i⁻¹β_{K,s,t} is an isomorphism (Tag 0GJU) — is an isomorphism.

The objects named in the statement, all defined in this library with no hypothesis beyond their data:

• smallEtaleDerivedCategory X (EtaleCohomology/DerivedEtale.lean) is D(X_étale), the derived category of abelian sheaves on the small étale site; K.IsGE m says K is bounded below.
• IsLocallyAcyclicRelative B hX hS f K (LocalAcyclicity/Basic.lean) is part (2) of Definition 0GJP: at every geometric point x of X and every geometric point t of Spec 𝒪^{sh}_{S,f(x)} the map α_{K,x,t} : K_x ⟶ RΓ(F_{x,t}, K) to the cohomology of the variety of vanishing cycles is an isomorphism; B is a big étale site containing X and S, a size device only.
• Stacks.Torsion.torsionCohomologyProperty (CategoryTheory/Sites/Cohomology/Torsion/ StacksFrontier/CategoryTheory/Sites/Cohomology/Torsion/DerivedCategoryOfTorsionSheaves.lean) says every cohomology sheaf of K is torsion.
• smallEtaleDerivedPushforwardAb f (EtaleCohomology/StalkOfDerivedPushforward.lean) is Rf_*, the total right derived functor of direct image; its existence is the proved instance Stacks.EtaleCohomology.hasRightDerivedFunctor_smallEtalePushforwardAb (EtaleCohomology/SmallEtaleUnboundedDerivations.lean), and that of RΓ the proved instance Stacks.EtaleCohomology.smallEtaleGlobalSectionsAb_hasRightDerivedFunctor (EtaleCohomology/SmallEtaleDerivedGlobalSectionsExistence.lean).
• smallEtaleDerivedSpecializationStalkAbAt B S hS s t (EtaleCohomology/SpecializationMapAndFibres.lean) is the specialization map sp : (−)_s ⟶ (−)_t on derived stalks, at the image of t in S.
• cospecializationUnit B hS f s t K (LocalAcyclicity/CospecializationMap.lean) is β_{K,s,t} : K|_Y ⟶ Rh_*(K|_{X_t}), and geometricFibreToStrictHenselizationBaseChange B hS f s (EtaleCohomology/StalkOfDerivedPushforward.lean) is i : X_s ⟶ Y.
• cospecializationMap B hS f s t K (LocalAcyclicity/CospecializationMapDerived.lean) is the cospecialization map of Section 0GJT.

theorem isIso_specializationStalk_and_cospecializationMap_of_isProper [IsProper f]
    (K : smallEtaleDerivedCategory X) (m : ℤ) (hK : K.IsGE m)
    (hla : IsLocallyAcyclicRelative B hX hS f K)
    (hKt : Stacks.Torsion.torsionCohomologyProperty.{u + 1} (smallEtaleTopology X) K) :
    IsIso ((smallEtaleDerivedSpecializationStalkAbAt B S hS s t).app
        ((smallEtaleDerivedPushforwardAb f).obj K)) ∧
      ∃ _ : IsIso ((smallEtaleDerivedPullbackAb
          (geometricFibreToStrictHenselizationBaseChange B hS f s)).map
            (cospecializationUnit B hS f s t K)),
        IsIso (cospecializationMap B hS f s t K)

Tag 0GKD, a lemma

Tag 0GKD in the Stacks Project.

AlgebraicGeometry.isIso_specializationMap_of_smooth_of_isProper_unconditional, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/LocalAcyclicity/SmoothProperSpecializationUnconditional.lean, line 68. References: SGA 4 XVI (2.3), “special case”; SGA 4½ Th finitude (5.2.4), “partial”; SGA 7 I (2.1), “more general”.

Let f : X ⟶ S be a morphism of schemes and F an abelian sheaf on X_étale. Assume

1. f is smooth and proper,
2. F is locally constant, and
3. for every geometric point x̄ of X the stalk F_{x̄} is a torsion group all of whose elements have order prime to the residue characteristic of x̄.

Then for every geometric point s̄ of S and every geometric point t̄ of Spec 𝒪^{sh}_{S,s̄} the specialization map sp : (Rf_*F)_{s̄} ⟶ (Rf_*F)_{t̄} is an isomorphism; the stalk at t̄ is taken at its image in S.

The strict henselization is formed in a big étale site B containing X and S, which exists for every pair of schemes; the statement holds for every such B.

theorem isIso_specializationMap_of_smooth_of_isProper_unconditional (B : BigEtaleSite.{u})
    {X S : Scheme.{u}} (hX : B.Contains X) (hS : B.Contains S) (f : X ⟶ S) [Smooth f]
    [IsProper f] (F : Sheaf X.smallEtaleTopology Ab.{u + 1})
    (hFlc : Stacks.ConstantSheaf.IsLocallyConstant X.smallEtaleTopology F)
    (hFtors : ∀ (x : GeometricPoint X) (a : (smallEtaleStalkAb x.hom).obj F),
      ∃ n : ℕ, 0 < n ∧
        (ringChar x.residueField = 0 ∨ Nat.Coprime n (ringChar x.residueField)) ∧ n • a = 0)
    (s : GeometricPoint S) (t : GeometricPoint (strictHenselization B S hS s)) :
    IsIso ((smallEtaleDerivedSpecializationStalkAbAt B S hS s t).app
      ((smallEtaleDerivedPushforwardAb f).obj
        ((DerivedCategory.singleFunctor (Sheaf X.smallEtaleTopology Ab.{u + 1}) 0).obj F)))

Sources

• Section 59.94, Tag 0GJT
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.isIso_pullback_cospecializationUnit · AlgebraicGeometry.isIso_pullback_cospecializationUnit_of_isGE · AlgebraicGeometry.specializationStalk_comp_baseChange_comp_cospecializationMap_of_isGE · AlgebraicGeometry.isIso_specializationStalk_and_cospecializationMap_of_isProper · AlgebraicGeometry.isIso_specializationMap_of_smooth_of_isProper_unconditional

Section59.95 Cohomological dimension#

Section 59.95 of the Stacks Project, Tag 0F0P. 8 tags here are carried by 9 declarations of the repository (all in StacksFrontier) and 1 registered declaration, quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F0Q, a definition

Tag 0F0Q in the Stacks Project.

AlgebraicGeometry.Scheme.etaleCohomologicalDimension, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension.lean, line 103.

Stacks Project, Tag 0F0Q. The least degree above which the étale cohomology of every torsion abelian sheaf on a quasi-compact, quasi-separated scheme vanishes.

public def etaleCohomologicalDimension (X : Scheme.{u}) [CompactSpace X]
    [QuasiSeparatedSpace X]
    [HasSheafify X.smallEtaleTopology AddCommGrpCat.{u + 1}]
    [HasExt.{u + 1} (Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})] : ℕ∞

Tag 0F0R, a lemma

Tag 0F0R in the Stacks Project.

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_iSup_of_isLimit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimensionLimits.lean, line 199. The attribute's comment: “the hypothesis h09YQ of this declaration is the transport of vanishing that Tags 0EYN and 09YQ supply: a degree in which the étale cohomology of every direct image f_{i,*}𝓕 vanishes is a degree in which that of 𝓕 vanishes”.

Stacks Project, Tag 0F0R (Étale Cohomology, Lemma 59.95.2). Let X = lim X_i be a directed limit of a system of quasi-compact and quasi-separated schemes with affine transition morphisms. Then cd(X) ≤ max cd(X_i).

The directed system is the cofiltered diagram D with affine maps and quasi-compact, quasi-separated values, the limit is the cone c with hc : IsLimit c, the projections f_i : X ⟶ X_i are c.π.app i, and the maximum over the index set is the supremum in the complete lattice ℕ∞, which is attained because a cofiltered category is nonempty.

h09YQ is the consequence of [Stacks, Tag 0EYN] and [Stacks, Tag 09YQ] that the proof uses: since 𝓕 = colim f_i^{-1} f_{i,*}𝓕 and hence H^q_étale(X, 𝓕) = colim H^q_étale(X_i, f_{i,*}𝓕), a degree in which the cohomology of every f_{i,*}𝓕 vanishes is a degree in which the cohomology of 𝓕 vanishes.

The proof is then the source's: a degree above ⨆ i, cd(X_i) is above every cd(X_i), so each H^q_étale(X_i, f_{i,*}𝓕) vanishes by the defining property of cd(X_i) applied to the sheaf f_{i,*}𝓕, which is torsion by [Stacks, Tag 0DDD] part (1) in degree zero — Stacks.EtaleCohomology.torsionSheafProperty_rightDerived_smallEtaleDirectImageAb read through CategoryTheory.Functor.rightDerivedZeroIsoSelf, the projection f_i being affine and hence quasi-compact and quasi-separated — and h09YQ transports the vanishing to X.

public theorem etaleCohomologicalDimension_le_iSup_of_isLimit
    [CompactSpace c.pt] [QuasiSeparatedSpace c.pt]
    [HasSheafify c.pt.smallEtaleTopology AddCommGrpCat.{u + 1}]
    [HasExt.{u + 1} (Sheaf c.pt.smallEtaleTopology AddCommGrpCat.{u + 1})]
    [∀ i : I, HasSheafify (D.obj i).smallEtaleTopology AddCommGrpCat.{u + 1}]
    [∀ i : I, HasExt.{u + 1} (Sheaf (D.obj i).smallEtaleTopology AddCommGrpCat.{u + 1})]
    (hc : IsLimit c)
    (h09YQ : ∀ F : Sheaf c.pt.smallEtaleTopology AddCommGrpCat.{u + 1},
      Stacks.Torsion.torsionSheafProperty c.pt.smallEtaleTopology F →
        ∀ q : ℕ, (∀ i : I, Subsingleton (Sheaf.H.{u + 1, u + 1}
            ((Stacks.EtaleCohomology.smallEtalePushforward
              (c.π.app i) AddCommGrpCat.{u + 1}).obj F) q)) →
          Subsingleton (Sheaf.H.{u + 1, u + 1} F q)) :
    etaleCohomologicalDimension c.pt ≤
      ⨆ i : I, etaleCohomologicalDimension (D.obj i)

Declarations registered for the tag:

• AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_iSup_of_isLimit_of_isDirected, declared in the repository, registered with add_to_stacks in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimensionDirectedLimits.lean, line 29 (StacksFrontier): “cd(lim X_i) ≤ max cd(X_i) for a directed limit of qcqs schemes with affine transition morphisms”.

Tag 0F0S, a lemma

Tag 0F0S in the Stacks Project.

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_one_add, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/AffineCurve.lean, line 237. The attribute's comment: “for X of dimension at most one, which contains the source's dimension one”. References: SGA 4 X (5.2), “special case”; SGA 4 XIV (3.2), “special case”; SGA 4½ Arcata (4.6.4), “special case”; SGA 5 I (1.5.2), “special case”.

Stacks Project, Tag 0F0S. Let K be a field and X an affine scheme of finite type over K of dimension at most one. Then cd(X) ≤ 1 + cd(K).

The source's statement is for X of dimension one; dimension at most one is no harder, and contains it.

The proof is the source's. Let 𝓕 be a torsion sheaf on X_étale. By the Leray spectral sequence for g : X ⟶ Spec K (Stacks.EtaleCohomology.LerayVanishing) it suffices that H^p(Spec K, R^qg_*𝓕) = 0 for p + q > 1 + cd(K). For q ≤ 1 this is the definition of cd(K), R^qg_*𝓕 being torsion (Stacks 0DDD). For q ≥ 2 the sheaf R^qg_*𝓕 vanishes: its stalk at a geometric point is the cohomology of the base change of X to the strict henselization, which is a separable closure Ω of K (Stacks 03Q9), and X_Ω is an affine curve over a separably closed field, of cohomological dimension at most one by etaleCohomologicalDimension_le_one_of_isSepClosed.

public theorem etaleCohomologicalDimension_le_one_add
    {K : Type u} [Field K] {X : Scheme.{u}} [IsAffine X] (g : X ⟶ Spec (.of K))
    [LocallyOfFiniteType g] (hX : topologicalKrullDim X ≤ 1) :
    etaleCohomologicalDimension X ≤ 1 + etaleCohomologicalDimension (Spec (.of K))

Tag 0F0T, a lemma

Tag 0F0T in the Stacks Project.

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_spec_le_add_trdeg, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/FieldExtension.lean, line 224.

Stacks Project, Tag 0F0T (Étale Cohomology, Lemma 59.95.4). Let L/K be a field extension. Then cd(L) ≤ cd(K) + trdeg_K(L).

The cohomological dimension of a field is that of its spectrum (Tag 0F0Q: "if X = Spec(A) we sometimes call this the cohomological dimension of A"), etaleCohomologicalDimension, valued in ℕ∞; the transcendence degree is the cardinal Algebra.trdeg K L read in ℕ∞ by Cardinal.toENat, so that an infinite transcendence degree is ⊤ and the inequality is then clear, as in the source.

The proof is the source's: a tower of transcendence degree one steps (etaleCohomologicalDimension_spec_le_add_trdeg_of_le_one_add, which also treats the algebraic case r = 0) reduces to trdeg_K(L) ≤ 1, which is etaleCohomologicalDimension_spec_le_one_add_of_trdeg_le_one (Tags 0F0R and 0F0S).

theorem etaleCohomologicalDimension_spec_le_add_trdeg
    (K L : Type u) [Field K] [Field L] [Algebra K L] :
    etaleCohomologicalDimension (Spec (.of L)) ≤
      etaleCohomologicalDimension (Spec (.of K)) + (Algebra.trdeg K L).toENat

Tag 0F0U, a lemma

Tag 0F0U in the Stacks Project.

AlgebraicGeometry.exists_strictHenselization_map_of_finiteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Morphisms/EtaleNeighbourhood/StrictHenselizationOverField.lean, line 563.

Stacks Project, Tag 0F0U. At a point x of a finite-type K-scheme, a strict henselization of the local ring admits a map from a separable closure of a rational function field in trdeg K κ(x) variables. The induced residue-field extension is purely inseparable, and the strict henselization is a filtered colimit of finite-type algebras of dimension at most dimₓ(X) - trdeg K κ(x).

theorem exists_strictHenselization_map_of_finiteType
    (K : Type u) [Field K] {X : Scheme.{u}} (f : X ⟶ Spec (.of K))
    [LocallyOfFiniteType f] [QuasiCompact f] (x : X) :
    letI := (((Scheme.ΓSpecIso (.of K)).inv ≫ f.appTop ≫
      X.presheaf.germ ⊤ x (by simp)).hom).toAlgebra
    letI := (((Scheme.ΓSpecIso (.of K)).inv ≫
      (Spec (.of K)).presheaf.germ ⊤ (f x) trivial ≫
      (Spec (.of K)).residue (f x) ≫ f.residueFieldMap x).hom).toAlgebra
    ∃ (ι E Rsh L : Type u)
      (_ : Field E) (_ : Algebra K E) (_ : Algebra (MvRatFunc ι K) E)
      (_ : IsScalarTower K (MvRatFunc ι K) E)
      (_ : IsSepClosure (MvRatFunc ι K) E)
      (_ : CommRing Rsh) (_ : IsLocalRing Rsh)
      (_ : Algebra K Rsh) (_ : Algebra (X.presheaf.stalk x) Rsh)
      (_ : IsScalarTower K (X.presheaf.stalk x) Rsh)
      (_ : Field L) (_ : Algebra (ResidueField (X.presheaf.stalk x)) L)
      (_ : IsSepClosure (ResidueField (X.presheaf.stalk x)) L),
      Cardinal.mk ι = Algebra.trdeg K (X.residueField x) ∧
      IsLocalRing.IsHenselization (X.presheaf.stalk x) L Rsh ∧
      ∃ e : E →ₐ[K] Rsh,
        letI := ((residue Rsh).comp e.toRingHom).toAlgebra
        IsPurelyInseparable E (ResidueField Rsh) ∧
          letI := e.toAlgebra
          IsFilteredColimitOfFiniteTypeAlgebrasWithDimensionAddLE E Rsh
            (Cardinal.toENat (Cardinal.mk ι)) (topologicalKrullDimAt X x)

Tag 0F0V, a proposition

Tag 0F0V in the Stacks Project.

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_add_of_topologicalKrullDim_le, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/AffineFiniteType.lean, line 142. The attribute's comment: “for every natural number n ≥ dim X, which also covers the empty scheme; the literal inequality cd(X) ≤ dim(X) + cd(K) for nonempty X is etaleCohomologicalDimension_le_topologicalKrullDim_add”. References: SGA 4 X (5.2), “special case”; SGA 4 XIV (3.2), “more general”.

Stacks Project, Tag 0F0V, for every bound n on the dimension. Let K be a field and let X be an affine scheme of finite type over K with dim(X) ≤ n. Then cd(X) ≤ n + cd(K).

The proof is the source's induction on the dimension: dimension at most one is Stacks 0F0S, and in dimension d ≥ 2 Noether normalization reduces to 𝔸^d_K, which is the inductive step etaleCohomologicalDimension_spec_polynomial_le. This form holds for every X, the empty one included; with n = dim(X) it is the source's cd(X) ≤ dim(X) + cd(K).

theorem etaleCohomologicalDimension_le_add_of_topologicalKrullDim_le
    (K : Type u) [Field K] (X : Scheme.{u}) (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f]
    [IsAffine X] (n : ℕ) (hX : topologicalKrullDim X ≤ n) :
    etaleCohomologicalDimension X ≤ n + etaleCohomologicalDimension (Spec (.of K))

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_topologicalKrullDim_add, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/AffineFiniteType.lean, line 197. The attribute's comment: “verbatim, cd(X) ≤ dim(X) + cd(K) in WithBot ℕ∞, for nonempty X: for X = ∅ the source's inequality reads 0 ≤ −∞ and fails; etaleCohomologicalDimension_le_add_of_topologicalKrullDim_le covers every X”. References: SGA 4 X (5.2), “special case”; SGA 5 VII (7.1.1), “a), special case”.

Stacks Project, Tag 0F0V. Let K be a field and let X be a nonempty affine scheme of finite type over K. Then cd(X) ≤ dim(X) + cd(K).

The inequality is read in WithBot ℕ∞, where dim(X) lives. Nonemptiness is needed: for X = ∅ the dimension is −∞ and cd(∅) = 0, so the inequality fails; the empty case is covered by etaleCohomologicalDimension_le_add_of_topologicalKrullDim_le.

theorem etaleCohomologicalDimension_le_topologicalKrullDim_add
    (K : Type u) [Field K] (X : Scheme.{u}) (f : X ⟶ Spec (.of K)) [LocallyOfFiniteType f]
    [IsAffine X] [Nonempty X] :
    (etaleCohomologicalDimension X : WithBot ℕ∞) ≤
      topologicalKrullDim X + (etaleCohomologicalDimension (Spec (.of K)) : WithBot ℕ∞)

Tag 0F0W, a lemma

Tag 0F0W in the Stacks Project.

AlgebraicGeometry.subsingleton_H_of_smallEtaleSupport_subset_trdegLESet, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/TrdegSupportVanishingUnconditional.lean, line 52. References: SGA 2 XIV (3.1), “special case”; SGA 4½ Arcata (4.6.4), “more general”.

Stacks Project, Tag 0F0W. Let K be a field and X an affine scheme of finite type over K. Let E_a ⊆ X be the set of points x with trdeg_K κ(x) ≤ a, and let 𝓕 be an abelian torsion sheaf on X_ét whose support is contained in E_a. Then H^b_ét(X, 𝓕) = 0 for b > a + cd(K).

The scheme is of finite type over K through f, the residue fields are K-algebras through f, E_a is AlgebraicGeometry.Scheme.trdegLESet K f a, the support is Stacks.EtaleCohomology.smallEtaleSupport (the points over which some geometric stalk is nonzero), torsion is Stacks.Torsion.torsionSheafProperty (Stacks 0DD7), cd(K) is the cohomological dimension AlgebraicGeometry.Scheme.etaleCohomologicalDimension of Spec K (Stacks 0F0Q), and the vanishing of H^b is Subsingleton.

theorem subsingleton_H_of_smallEtaleSupport_subset_trdegLESet (K : Type u) [Field K]
    {X : Scheme.{u}} (f : X ⟶ Spec (.of K)) [IsAffine X] [LocallyOfFiniteType f] (a : ℕ)
    (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})
    (hF : Stacks.Torsion.torsionSheafProperty X.smallEtaleTopology F)
    (hsupp : Stacks.EtaleCohomology.smallEtaleSupport F ⊆ Scheme.trdegLESet K f a)
    (b : ℕ) (hb : (a : ℕ∞) + Scheme.etaleCohomologicalDimension (Spec (.of K)) < b) :
    Subsingleton (Sheaf.H.{u + 1, u + 1} F b)

Tag 0F0X, a lemma

Tag 0F0X in the Stacks Project.

AlgebraicGeometry.smallEtaleSupport_rightDerived_etaleAbelianDirectImage_subset, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/AffineHigherDirectImageSupport.lean, line 466. References: SGA 4 XIV (3.1), “more general”.

Stacks Project, Tag 0F0X. Let f : X ⟶ Y be an affine morphism of schemes locally of finite type over a field K (the structure morphisms gX, gY, with f ≫ gY = gX), let E_a(X) be the set of points x ∈ X with trdeg_K κ(x) ≤ a (AlgebraicGeometry.Scheme.trdegLESet), and let 𝓕 be an abelian torsion sheaf on X_ét whose support is contained in E_a(X). Then R^q f_* 𝓕 has support contained in E_{a - q}(Y): every point y of its support has trdeg_K κ(y) + q ≤ a (so the support is empty when q > a).

The proof is the source's: Stacks 03Q9 (AlgebraicGeometry.isZero_stalk_rightDerived_iff_subsingleton_sheafH) computes the stalk at a geometric point over y as the cohomology of X ×_Y Spec(𝒪^{sh}_{Y,y}), which vanishes by AlgebraicGeometry.subsingleton_H_strictHenselizationBaseChange_of_trdeg (Stacks 0F0U, 0F0W over a separable closure of K(t_1, …, t_b), and the vanishing form of Stacks 03Q6).

theorem smallEtaleSupport_rightDerived_etaleAbelianDirectImage_subset
    (K : Type u) [Field K] {X Y : Scheme.{u}} (gX : X ⟶ Spec (.of K)) (gY : Y ⟶ Spec (.of K))
    [LocallyOfFiniteType gX] [LocallyOfFiniteType gY] (f : X ⟶ Y) [IsAffineHom f]
    (hfg : f ≫ gY = gX) (a : ℕ) (F : Sheaf X.smallEtaleTopology AddCommGrpCat.{u + 1})
    (hF : Stacks.Torsion.torsionSheafProperty X.smallEtaleTopology F)
    (hsupp : smallEtaleSupport F ⊆ Scheme.trdegLESet K gX a) (q : ℕ) :
    smallEtaleSupport (((etaleAbelianDirectImage.{u, u + 1} f).rightDerived q).obj F) ⊆
      {y | letI : Algebra K (Y.residueField y) :=
          ((Scheme.ΓSpecIso (.of K)).inv ≫ gY.appTop ≫ Y.Γevaluation y).hom.toAlgebra
        Algebra.trdeg K (Y.residueField y) + q ≤ a}

Sources

• Section 59.95, Tag 0F0P
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Scheme.etaleCohomologicalDimension · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_iSup_of_isLimit · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_iSup_of_isLimit_of_isDirected · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_one_add · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_spec_le_add_trdeg · AlgebraicGeometry.exists_strictHenselization_map_of_finiteType · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_add_of_topologicalKrullDim_le · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_le_topologicalKrullDim_add

Section59.96 Finite cohomological dimension#

Section 59.96 of the Stacks Project, Tag 0F0Y. 6 tags here are carried by 12 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F0Z, a definition

Tag 0F0Z in the Stacks Project.

AlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension.lean, line 163.

Stacks Project, Tag 0F0Z. The cohomological dimension cd(f) of a quasi-compact, quasi-separated morphism of schemes f : X ⟶ Y: the smallest d ∈ {0, 1, 2, …} ∪ {∞} such that R^i f_* F = 0 for every abelian torsion sheaf F on X_ét and every i > d.

public def etaleCohomologicalDimension [QuasiCompact f] [QuasiSeparated f] : ℕ∞

Tag 0F10, a lemma

Tag 0F10 in the Stacks Project.

AlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension_lt_top_of_finiteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/FiniteType.lean, line 92. The attribute's comment: “(1): cd(f) < ∞ for a morphism of schemes of finite type over a field”.

Stacks Project, Tag 0F10, part (1). A morphism f : X ⟶ Y of schemes of finite type over a field K has finite cohomological dimension: cd(f) < ∞.

theorem Hom.etaleCohomologicalDimension_lt_top_of_finiteType (K : Type u) [Field K]
    {X Y : Scheme.{u}} (x : X ⟶ Spec (.of K)) [LocallyOfFiniteType x] [QuasiCompact x]
    (y : Y ⟶ Spec (.of K)) [LocallyOfFiniteType y] [QuasiCompact y] (f : X ⟶ Y)
    (hf : f ≫ y = x) :
    haveI := quasiCompact_of_finiteType K x y f
    haveI := quasiSeparated_of_finiteType K x f
    f.etaleCohomologicalDimension < ⊤

AlgebraicGeometry.Scheme.etaleCohomologicalDimension_lt_top_of_finiteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimension/FiniteTypeAbsolute.lean, line 48. The attribute's comment: “(2): cd(X) < ∞ for X of finite type over a field K with cd(K) < ∞”.

Stacks Project, Tag 0F10, part (2). If the field K has finite cohomological dimension cd(K) = cd(Spec K) < ∞, then every scheme X of finite type over K has finite cohomological dimension: cd(X) < ∞.

theorem etaleCohomologicalDimension_lt_top_of_finiteType (K : Type u) [Field K]
    {X : Scheme.{u}} (x : X ⟶ Spec (.of K)) [LocallyOfFiniteType x] [QuasiCompact x]
    (hK : etaleCohomologicalDimension (Spec (.of K)) < ⊤) :
    haveI : CompactSpace X := compactSpace_of_finiteType K x
    haveI : QuasiSeparatedSpace X := quasiSeparatedSpace_of_finiteType K x
    etaleCohomologicalDimension X < ⊤

Tag 0F11, a lemma

Tag 0F11 in the Stacks Project.

Stacks.EtaleCohomology.preservesColimitsOfShape_of_finiteCohomologicalDimension_directImage, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimensionDirectSums.lean, line 106. The attribute's comment: “(1) Rf_ : D(X_étale, Λ) ⟶ D(Y_étale, Λ) commutes with direct sums when f_ has finite cohomological dimension. Tag 09Z1, that each R^q f_* commutes with direct sums, is carried as the hypothesis hsums, since it is not proved in this library; for Λ a general ring rather than only ℤ/nℤ.”.

[Stacks, Tag 0F11] (1). Let Λ be a ring and f : X ⟶ Y a morphism of schemes such that the direct image of sheaves of Λ-modules on the small étale sites has finite cohomological dimension, i.e. there is a degree d above which every higher direct image R^q f_* vanishes on every sheaf of Λ-modules. Then any unbounded right derived functor Rf_* : D(X_étale, Λ) ⟶ D(Y_étale, Λ) commutes with direct sums.

The hypothesis hcd is the source's cd(f) < ∞, in the shape its proof uses it and in which [Stacks, Tag 095U] supplies it. The hypothesis hsums is [Stacks, Tag 09Z1], that each R^q f_* commutes with direct sums; it is carried because that tag is not proved in this library, and nothing else here is assumed.

theorem preservesColimitsOfShape_of_finiteCohomologicalDimension_directImage
    (Λ : Type r) [Ring Λ] {X Y : Scheme.{u}} (f : X ⟶ Y)
    [HasDerivedCategory.{z} (EtaleModules.{u, r} Λ X)]
    [HasDerivedCategory.{z} (EtaleModules.{u, r} Λ Y)]
    (hcd : ∃ d : ℕ, ∀ q : ℕ, d ≤ q → ∀ F : EtaleModules.{u, r} Λ X,
      IsZero (((etaleModuleDirectImage.{u, r} Λ f).rightDerived q).obj F))
    (hsums : ∀ (q : ℕ) (I : Type (max (u + 1) r)),
      PreservesColimitsOfShape (Discrete I)
        ((etaleModuleDirectImage.{u, r} Λ f).rightDerived q))
    (RF : DerivedCategory (EtaleModules.{u, r} Λ X) ⥤
      DerivedCategory (EtaleModules.{u, r} Λ Y))
    (α : (etaleModuleDirectImage.{u, r} Λ f).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, r} Λ X) (ComplexShape.up ℤ))]
    (I : Type (max (u + 1) r)) :
    PreservesColimitsOfShape (Discrete I) RF

Stacks.EtaleCohomology.preservesColimitsOfShape_of_finiteCohomologicalDimension_globalSections, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/CohomologicalDimensionDirectSums.lean, line 139. The attribute's comment: “(2) RΓ(X, -) : D(X_étale, Λ) ⟶ D(Λ) commutes with direct sums when Γ has finite cohomological dimension. Tag 09Z1, that each H^q(X, -) commutes with direct sums, is carried as the hypothesis hsums, since it is not proved in this library; for Λ a general ring rather than only ℤ/nℤ.”.

[Stacks, Tag 0F11] (2). Let Λ be a ring and X a scheme such that global sections of sheaves of Λ-modules on the small étale site of X has finite cohomological dimension, i.e. there is a degree d above which every H^q(X, -) vanishes on every sheaf of Λ-modules. Then any unbounded right derived functor RΓ(X, -) : D(X_étale, Λ) ⟶ D(Λ) commutes with direct sums.

The hypothesis hcd is the source's cd(X) < ∞, and hsums is [Stacks, Tag 09Z1] read on the structure morphism of X, that each H^q(X, -) commutes with direct sums; it is carried because that tag is not proved in this library.

theorem preservesColimitsOfShape_of_finiteCohomologicalDimension_globalSections
    (Λ : Type r) [Ring Λ] (X : Scheme.{u})
    [HasDerivedCategory.{z} (EtaleModules.{u, r} Λ X)]
    [HasDerivedCategory.{z} (ModuleCat.{max (u + 1) r} Λ)]
    (hcd : ∃ d : ℕ, ∀ q : ℕ, d ≤ q → ∀ F : EtaleModules.{u, r} Λ X,
      IsZero (((etaleModuleGlobalSections.{u, r} Λ X).rightDerived q).obj F))
    (hsums : ∀ (q : ℕ) (I : Type (max (u + 1) r)),
      PreservesColimitsOfShape (Discrete I)
        ((etaleModuleGlobalSections.{u, r} Λ X).rightDerived q))
    (RF : DerivedCategory (EtaleModules.{u, r} Λ X) ⥤
      DerivedCategory (ModuleCat.{max (u + 1) r} Λ))
    (α : (etaleModuleGlobalSections.{u, r} Λ X).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, r} Λ X) (ComplexShape.up ℤ))]
    (I : Type (max (u + 1) r)) :
    PreservesColimitsOfShape (Discrete I) RF

Stacks.EtaleCohomology.preservesColimitsOfShape_derivedDirectImage_zmod, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/TorsionCoefficientsDirectSums.lean, line 422. The attribute's comment: “(1): Rf_* : D(X_étale, ℤ/nℤ) ⟶ D(Y_étale, ℤ/nℤ) commutes with direct sums for f quasi-compact and quasi-separated with cd(f) < ∞, n ≥ 1”.

[Stacks, Tag 0F11] (1). Let n ≥ 1 and let f : X ⟶ Y be a quasi-compact and quasi-separated morphism of schemes with cd(f) < ∞. Then the unbounded derived direct image Rf_* : D(X_étale, ℤ/nℤ) ⟶ D(Y_étale, ℤ/nℤ) commutes with direct sums.

Here cd(f) is AlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension, Tag 0F0Z, and Rf_* is any unbounded right derived functor of the direct image of sheaves of ℤ/nℤ-modules. The two inputs of the argument of the source are proved: the uniform vanishing of the higher direct images on sheaves of ℤ/nℤ-modules is Stacks.EtaleCohomology.exists_isZero_rightDerived_etaleModuleDirectImage_zmod, and their commuting with direct sums (Tag 09Z1 for a constant system) is Stacks.EtaleCohomology.preservesColimitsOfShape_discrete_rightDerived_etaleModuleDirectImage.

theorem preservesColimitsOfShape_derivedDirectImage_zmod
    {X Y : Scheme.{u}} (f : X ⟶ Y) [QuasiCompact f] [QuasiSeparated f]
    (hcd : f.etaleCohomologicalDimension < ⊤) (n : ℕ) (hn : 0 < n)
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} (ZMod n) X)]
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} (ZMod n) Y)]
    (RF : DerivedCategory (EtaleModules.{u, 0} (ZMod n) X) ⥤
      DerivedCategory (EtaleModules.{u, 0} (ZMod n) Y))
    (α : (etaleModuleDirectImage.{u, 0} (ZMod n) f).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, 0} (ZMod n) X) (ComplexShape.up ℤ))]
    (I : Type (u + 1)) :
    PreservesColimitsOfShape (Discrete I) RF

Stacks.EtaleCohomology.preservesColimitsOfShape_derivedGlobalSections_zmod, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/TorsionCoefficientsDirectSums.lean, line 498. The attribute's comment: “(2): RΓ(X, -) : D(X_étale, ℤ/nℤ) ⟶ D(ℤ/nℤ) commutes with direct sums for X quasi-compact and quasi-separated with cd(X) < ∞, n ≥ 1”.

[Stacks, Tag 0F11] (2). Let n ≥ 1 and let X be a quasi-compact and quasi-separated scheme with cd(X) < ∞. Then the unbounded derived global sections RΓ(X, -) : D(X_étale, ℤ/nℤ) ⟶ D(ℤ/nℤ) commute with direct sums.

Here cd(X) is AlgebraicGeometry.Scheme.etaleCohomologicalDimension, Tag 0F0Q, and RΓ(X, -) is any unbounded right derived functor of global sections of sheaves of ℤ/nℤ-modules. The two inputs of the argument of the source are proved: the uniform vanishing of the cohomology of sheaves of ℤ/nℤ-modules is Stacks.EtaleCohomology.exists_isZero_rightDerived_etaleModuleGlobalSections_zmod, and its commuting with direct sums is preservesColimitsOfShape_discrete_rightDerived_etaleModuleGlobalSections.

theorem preservesColimitsOfShape_derivedGlobalSections_zmod
    (hcd : Scheme.etaleCohomologicalDimension X < ⊤)
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} (ZMod n) X)]
    [HasDerivedCategory.{z} (ModuleCat.{u + 1} (ZMod n))]
    (RF : DerivedCategory (EtaleModules.{u, 0} (ZMod n) X) ⥤
      DerivedCategory (ModuleCat.{u + 1} (ZMod n)))
    (α : (etaleModuleGlobalSections.{u, 0} (ZMod n) X).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, 0} (ZMod n) X) (ComplexShape.up ℤ))]
    (I : Type (u + 1)) :
    PreservesColimitsOfShape (Discrete I) RF

Stacks.EtaleCohomology.preservesColimitsOfShape_derivedDirectImage_of_natCast_eq_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/TorsionCoefficientsDirectSums.lean, line 569. The attribute's comment: “(1)”.

[Stacks, Tag 0F11] (1). Let n ≥ 1 and let f : X ⟶ Y be a quasi-compact and quasi-separated morphism of schemes with cd(f) < ∞. Then Rf_* : D(X_étale, Λ) ⟶ D(Y_étale, Λ) commutes with direct sums, for every commutative ring Λ in which n = 0; the source's case is Λ = ℤ/nℤ.

Here cd(f) is AlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension, [Stacks, Tag 0F0Z], and Rf_* is any unbounded right derived functor RF of the direct image of sheaves of Λ-modules on the small étale sites, with derivation unit α, in any model of the two derived categories.

theorem preservesColimitsOfShape_derivedDirectImage_of_natCast_eq_zero
    {X Y : Scheme.{u}} (f : X ⟶ Y) [QuasiCompact f] [QuasiSeparated f]
    (hcd : f.etaleCohomologicalDimension < ⊤)
    (Λ : Type) [CommRing Λ] (n : ℕ) (hn : 0 < n) (hΛ : (n : Λ) = 0)
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} Λ X)]
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} Λ Y)]
    (RF : DerivedCategory (EtaleModules.{u, 0} Λ X) ⥤ DerivedCategory (EtaleModules.{u, 0} Λ Y))
    (α : (etaleModuleDirectImage.{u, 0} Λ f).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, 0} Λ X) (ComplexShape.up ℤ))]
    (I : Type (u + 1)) :
    PreservesColimitsOfShape (Discrete I) RF

Stacks.EtaleCohomology.preservesColimitsOfShape_derivedGlobalSections_of_natCast_eq_zero, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/TorsionCoefficientsDirectSums.lean, line 596. The attribute's comment: “(2)”.

[Stacks, Tag 0F11] (2). Let n ≥ 1 and let X be a quasi-compact and quasi-separated scheme with cd(X) < ∞. Then RΓ(X, -) : D(X_étale, Λ) ⟶ D(Λ) commutes with direct sums, for every commutative ring Λ in which n = 0; the source's case is Λ = ℤ/nℤ.

Here cd(X) is AlgebraicGeometry.Scheme.etaleCohomologicalDimension, [Stacks, Tag 0F0Q], and RΓ(X, -) is any unbounded right derived functor RF of the global sections of sheaves of Λ-modules on the small étale site, with derivation unit α, in any model of the two derived categories.

theorem preservesColimitsOfShape_derivedGlobalSections_of_natCast_eq_zero
    (X : Scheme.{u}) [CompactSpace X] [QuasiSeparatedSpace X]
    (hcd : Scheme.etaleCohomologicalDimension X < ⊤)
    (Λ : Type) [CommRing Λ] (n : ℕ) (hn : 0 < n) (hΛ : (n : Λ) = 0)
    [HasDerivedCategory.{z} (EtaleModules.{u, 0} Λ X)]
    [HasDerivedCategory.{z} (ModuleCat.{u + 1} Λ)]
    (RF : DerivedCategory (EtaleModules.{u, 0} Λ X) ⥤ DerivedCategory (ModuleCat.{u + 1} Λ))
    (α : (etaleModuleGlobalSections.{u, 0} Λ X).mapHomotopyCategory (ComplexShape.up ℤ) ⋙
      DerivedCategory.Qh ⟶ DerivedCategory.Qh ⋙ RF)
    [RF.IsRightDerivedFunctor α
      (HomotopyCategory.quasiIso (EtaleModules.{u, 0} Λ X) (ComplexShape.up ℤ))]
    (I : Type (u + 1)) :
    PreservesColimitsOfShape (Discrete I) RF

Tag 0F0D, a lemma

Tag 0F0D in the Stacks Project.

Stacks.EtaleCohomology.preservesColimitsOfShape_etaleModuleDerivedDirectImageUnbounded_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperDirectImageDirectSums.lean, line 274. The attribute's comment: “for a torsion ring Λ in any universe, which contains Λ = ℤ/nℤ, n ≥ 1”.

Let f : X ⟶ Y be a proper morphism of schemes and let Λ be a torsion ring, in any universe; for instance Λ = ℤ/nℤ with n ≥ 1. Then the unbounded derived direct image Rf_* : D(X_étale, Λ) ⟶ D(Y_étale, Λ) commutes with direct sums.

theorem preservesColimitsOfShape_etaleModuleDerivedDirectImageUnbounded_of_isProper
    {X Y : Scheme.{u}} (f : X ⟶ Y) [IsProper f]
    (hΛ : ∃ m : ℕ, 0 < m ∧ (m : Λ) = 0) (I : Type (max (u + 1) r)) :
    PreservesColimitsOfShape (Discrete I) (etaleModuleDerivedDirectImageUnbounded.{u, r} Λ f)

Tag 0F12, a lemma

Tag 0F12 in the Stacks Project.

Stacks.EtaleCohomology.ProjectionFormulaConstant.isIso_projectionFormulaMap_toPoint, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProjectionFormulaConstant/Main.lean, line 204. The attribute's comment: “the canonical map of Section 0943 for the structure morphism of (X_étale, Λ̲) to the point ringed by Λ, on the shared ringed topos EtaleCoefficients.ringedTopos; D(pt, Λ) is D(Λ) through EtaleCoefficients.pointDerivedEquivalence”.

Let X be a quasi-compact and quasi-separated scheme with cd(X) < ∞ and let Λ be a torsion ring. For E ∈ D(X_étale, Λ) and K ∈ D(pt, Λ) = D(Λ), the canonical map of Cohomology on Sites, Section 0943, for the structure morphism π : (X_étale, Λ̲) ⟶ (pt, Λ),

RΓ(X, E) ⊗^L_Λ K = Rπ_*E ⊗^L_Λ K ⟶ Rπ_*(E ⊗^L_Λ Lπ^*K) = RΓ(X, E ⊗^L_Λ K̲),

is an isomorphism.

theorem isIso_projectionFormulaMap_toPoint [CompactSpace X] [QuasiSeparatedSpace X]
    (hcd : Scheme.etaleCohomologicalDimension X < ⊤) (hΛ : IsAddTorsion Λ)
    (E : DerivedCategory (SheafOfModules.{u + 1}
      (Stacks.SheafOfModules.Tensor.ringSheaf (ringedTopos Λ X).O)))
    (K : DerivedCategory (SheafOfModules.{u + 1}
      (Stacks.SheafOfModules.Tensor.ringSheaf (pointRingedTopos Λ).O))) :
    IsIso (projectionFormulaMap (toPoint Λ X) E K)

Tag 0F0G, a lemma

Tag 0F0G in the Stacks Project.

Stacks.EtaleCohomology.isIso_etalePushforwardProjectionMap_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperProjectionFormula/TorsionProjectionFormula.lean, line 149.

[Stacks, Tag 0F0G]. Let f : X ⟶ Y be a proper morphism of schemes and Λ a torsion ring. For E ∈ D(X_étale, Λ) and K ∈ D(Y_étale, Λ), unbounded, the canonical map of [Stacks, Tag 0943] Rf_*E ⊗^L_Λ K ⟶ Rf_*(E ⊗^L_Λ f^{-1}K) is an isomorphism in D(Y_étale, Λ).

theorem isIso_etalePushforwardProjectionMap_of_isProper (hΛ : AddMonoid.IsTorsion Λ)
    {X Y : Scheme.{u}} (f : X ⟶ Y) [IsProper f]
    (E : DerivedCategory (EtaleModules.{u, 0} Λ X))
    (K : DerivedCategory (EtaleModules.{u, 0} Λ Y)) :
    IsIso (etalePushforwardProjectionMap.{u, 0} Λ f E K)

Sources

• Section 59.96, Tag 0F0Y
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension · AlgebraicGeometry.Scheme.Hom.etaleCohomologicalDimension_lt_top_of_finiteType · AlgebraicGeometry.Scheme.etaleCohomologicalDimension_lt_top_of_finiteType · Stacks.EtaleCohomology.preservesColimitsOfShape_of_finiteCohomologicalDimension_directImage · Stacks.EtaleCohomology.preservesColimitsOfShape_of_finiteCohomologicalDimension_globalSections · Stacks.EtaleCohomology.preservesColimitsOfShape_derivedDirectImage_zmod · Stacks.EtaleCohomology.preservesColimitsOfShape_derivedGlobalSections_zmod · Stacks.EtaleCohomology.preservesColimitsOfShape_derivedDirectImage_of_natCast_eq_zero

Section59.97 Künneth in étale cohomology#

Section 59.97 of the Stacks Project, Tag 0F13. 9 tags here are carried by 15 declarations of the repository (all in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F1E, a remark

Tag 0F1E in the Stacks Project.

Stacks.Cohomology.KunnethMapOnRingedTopoi.directImageMap, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Cohomology/DerivedTensor/KunnethMap.lean, line 174. The attribute's comment: “the canonical map Rf_E ⟶ Rc_p⁻¹E”.

The canonical map Rf_*E ⟶ Rc_*Lp^*E for a factorization c = f ∘ p: the adjoint transpose, for Lc^* ⊣ Rc_*, of pullbackDirectImageCounit.

def directImageMap (hp : f.comp p = c)
    (E : DerivedCategory (SheafOfModules.{u} (Stacks.SheafOfModules.Tensor.ringSheaf X.O))) :
    (derivedPushforward f).obj E ⟶
      (derivedPushforward c).obj ((derivedPullback p).obj E)

Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfCupProduct, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Cohomology/DerivedTensor/KunnethMap.lean, line 214. The attribute's comment: “the canonical map, via the relative cup product of Remark 0B6C”.

The Künneth map, via the relative cup product. For morphisms of ringed topoi p : W ⟶ X, f : X ⟶ S, q : W ⟶ Y, g : Y ⟶ S and c : W ⟶ S with c = f ∘ p = g ∘ q, and for E ∈ D(𝒪_X), K ∈ D(𝒪_Y), the canonical map

Rf_*E ⊗^L_{𝒪_S} Rg_*K ⟶ Rc_*(Lp^*E ⊗^L_{𝒪_W} Lq^*K).

It is the canonical maps Rf_*E ⟶ Rc_*Lp^*E and Rg_*K ⟶ Rc_*Lq^*K in the two tensor factors, followed by the relative cup product of [Stacks, Tag 0B6C] for c.

def kunnethMapOfCupProduct :
    ((derivedTensorProductFunctor S.O).obj ((derivedPushforward f).obj E)).obj
        ((derivedPushforward g).obj K) ⟶
      (derivedPushforward c).obj
        (((derivedTensorProductFunctor W.O).obj ((derivedPullback p).obj E)).obj
          ((derivedPullback q).obj K))

Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfAdjoint, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Cohomology/DerivedTensor/KunnethMap.lean, line 236. The attribute's comment: “the canonical map, as the adjoint of the map on inverse images”.

The Künneth map, as an adjoint. The same canonical map

Rf_*E ⊗^L_{𝒪_S} Rg_*K ⟶ Rc_*(Lp^*E ⊗^L_{𝒪_W} Lq^*K),

written as the adjoint, for Lc^* ⊣ Rc_*, of the compatibility of derived pullback with the derived tensor product ([Stacks, Tag 07A4]) followed by the maps Lc^*Rf_*E ⟶ Lp^*E and Lc^*Rg_*K ⟶ Lq^*K in the two factors.

def kunnethMapOfAdjoint :
    ((derivedTensorProductFunctor S.O).obj ((derivedPushforward f).obj E)).obj
        ((derivedPushforward g).obj K) ⟶
      (derivedPushforward c).obj
        (((derivedTensorProductFunctor W.O).obj ((derivedPullback p).obj E)).obj
          ((derivedPullback q).obj K))

Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfCupProduct_eq_kunnethMapOfAdjoint, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/CategoryTheory/Sites/Cohomology/DerivedTensor/KunnethMap.lean, line 252. The attribute's comment: “the two constructions of the remark agree”.

The two constructions of the Künneth map agree: kunnethMapOfCupProduct equals kunnethMapOfAdjoint.

theorem kunnethMapOfCupProduct_eq_kunnethMapOfAdjoint :
    kunnethMapOfCupProduct p f c hmosp hmosf hmosc q g hmosq hmosg hp hq E K =
      kunnethMapOfAdjoint p f c hmosp hmosf hmosc q g hmosq hmosg hp hq E K

Tag 0F14, a lemma

Tag 0F14 in the Stacks Project.

Stacks.EtaleCohomology.isIso_etaleKunnethCompositeMap_zmod_of_finiteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperKunneth/KunnethFiniteType.lean, line 68. The attribute's comment: “(2); on D((Spec k)_et, ℤ/nℤ), which global sections identify with D(ℤ/nℤ) for k separably closed, so that the direct images to Spec k are the derived global sections”. References: SGA 4½ Th finitude (1.11), “Künneth arrow RΓ(X”.

[Stacks, Tag 0F14] (2). Let k be a separably closed field, f : X ⟶ Spec k proper, g : Y ⟶ Spec k of finite type, and n ≥ 1. For E ∈ D(X_étale, ℤ/nℤ) and K ∈ D(Y_étale, ℤ/nℤ), unbounded, the Künneth map RΓ(X, E) ⊗^L\_\{ℤ/nℤ} RΓ(Y, K) = Rf\_*E ⊗^L Rg\_*K ⟶ R(pr₂ ≫ g)\_*(pr₁⁻¹E ⊗^L pr₂⁻¹K) = RΓ(X ×\_k Y, pr₁⁻¹E ⊗^L pr₂⁻¹K) in D((Spec k)\_étale, ℤ/nℤ) = D(ℤ/nℤ) is an isomorphism.

theorem isIso_etaleKunnethCompositeMap_zmod_of_finiteType (n : ℕ) (hn : 0 < n) (k : Type u)
    [Field k] [IsSepClosed k] {X Y : Scheme.{u}} (f : X ⟶ Spec (.of k)) [IsProper f]
    (g : Y ⟶ Spec (.of k)) [LocallyOfFiniteType g] [QuasiCompact g]
    (E : DerivedCategory (EtaleModules.{u, 0} (ZMod n) X))
    (K : DerivedCategory (EtaleModules.{u, 0} (ZMod n) Y)) :
    IsIso (etaleKunnethCompositeMap.{u, 0} (ZMod n) f g (pullback.fst f g) (pullback.snd f g)
      pullback.condition E K)

Stacks.EtaleCohomology.isIso_etaleKunnethCompositeMap_of_isGE_of_isProper, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ProperKunneth/KunnethIntegral.lean, line 69. The attribute's comment: “(1); on D((Spec k)_et, ℤ), which global sections identify with D(ℤ) for k separably closed, so that the direct images to Spec k are the derived global sections”.

[Stacks, Tag 0F14] (1). Let k be a separably closed field, f : X ⟶ Spec k proper and g : Y ⟶ Spec k with Y quasi-compact and quasi-separated. Let E ∈ D⁺(X_étale) have torsion cohomology sheaves and K ∈ D⁺(Y_étale). Then the Künneth map RΓ(X, E) ⊗^L\_ℤ RΓ(Y, K) = Rf\_*E ⊗^L\_ℤ Rg\_*K ⟶ R(pr₂ ≫ g)\_*(pr₁⁻¹E ⊗^L\_ℤ pr₂⁻¹K) = RΓ(X ×\_k Y, pr₁⁻¹E ⊗^L\_ℤ pr₂⁻¹K) in D((Spec k)\_étale, ℤ) = D(ℤ) is an isomorphism.

theorem isIso_etaleKunnethCompositeMap_of_isGE_of_isProper (k : Type u) [Field k]
    [IsSepClosed k] {X Y : Scheme.{u}} (f : X ⟶ Spec (.of k)) [IsProper f]
    (g : Y ⟶ Spec (.of k)) [CompactSpace Y] [QuasiSeparatedSpace Y]
    (E : DerivedCategory (EtaleModules.{u, 0} ℤ X)) (a : ℤ) (hE : E.IsGE a)
    (hT : ∀ n, IsTorsionEtaleModule.{u, 0} ℤ X ((DerivedCategory.homologyFunctor _ n).obj E))
    (K : DerivedCategory (EtaleModules.{u, 0} ℤ Y)) (b : ℤ) (hK : K.IsGE b) :
    IsIso (etaleKunnethCompositeMap.{u, 0} ℤ f g (pullback.fst f g) (pullback.snd f g)
      pullback.condition E K)

Tag 0F1F, a lemma

Tag 0F1F in the Stacks Project.

AlgebraicGeometry.subsingleton_sheafH_and_epi_counit_of_supportSheaf_subset_closedPoints, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ClosedPointSupport/Main.lean, line 75.

Let K be a separably closed field, let X be a scheme of finite type over K and let F be an abelian sheaf on the small étale site of X whose support is contained in the set of closed points of X. Then H^q(X, F) = 0 for q > 0, and F is globally generated: the constant sheaf on its group of global sections surjects onto it.

public theorem subsingleton_sheafH_and_epi_counit_of_supportSheaf_subset_closedPoints
    (B : BigEtaleSite.{u}) (K : Type u) [Field K] [IsSepClosed K] (X : B.Scheme)
    (f : X.obj ⟶ Spec (.of K)) [LocallyOfFiniteType f] [QuasiCompact f]
    (F : SmallEtaleAbelianSheaf B X)
    (hF : supportSheaf B X.obj X.property F ⊆ {x : X.obj | IsClosed ({x} : Set X.obj)}) :
    (∀ q : ℕ, 0 < q → Subsingleton (Sheaf.H.{u + 1, u + 1} F q)) ∧
      Epi ((constantSheafΓAdj (SmallEtale.topology B X.obj X.property)
        AddCommGrpCat.{u + 1}).counit.app F)

Tag 0F1G, a lemma

Tag 0F1G in the Stacks Project.

AlgebraicGeometry.isZero_iff_forall_isZero_etaleHypercohomology_of_closedPointSupport, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/ClosedPointSupportDetection.lean, line 587.

[Stacks, Tag 0F1G]. Let K be a separably closed field, X a scheme of finite type over K and Q an object of the unbounded derived category of abelian sheaves on the small étale site of X whose stalks vanish away from the closed points of X. Then Q is zero if and only if all of its hypercohomology groups vanish.

One direction is formal. For the other, Q is zero exactly when every cohomology sheaf H^q(Q) is zero; the stalk hypothesis says that each H^q(Q) is supported at closed points, so [Stacks, Tag 0BKZ] identifies H^q(X, Q) with Γ(X, H^q(Q)) (AlgebraicGeometry.nonempty_hypercohomologyAbIsoΓHomology_of_supportSheaf_subset_closedPoints), and [Stacks, Tag 0F1F] makes a sheaf supported at closed points with no global sections zero (AlgebraicGeometry.isZero_of_isZero_Γ_of_supportSheaf_subset_closedPoints).

public theorem isZero_iff_forall_isZero_etaleHypercohomology_of_closedPointSupport
    (K : Type u) [Field K] [IsSepClosed K] (f : X ⟶ Spec (.of K))
    [LocallyOfFiniteType f] [QuasiCompact f]
    (Q : DerivedCategory (Sheaf (SmallEtale.topology B X hX) AddCommGrpCat.{u + 1}))
    (hQ : ∀ (p : GeometricPoint X) (q : ℤ), ¬ IsClosed ({p.hom default} : Set X) →
      IsZero (etaleStalk B X hX p
        ((DerivedCategory.homologyFunctor
          (Sheaf (SmallEtale.topology B X hX) AddCommGrpCat.{u + 1}) q).obj Q).obj)) :
    IsZero Q ↔
      ∀ q : ℤ, IsZero ((DerivedCategory.homologyFunctor AddCommGrpCat.{u + 1} q).obj
        ((smallEtaleDerivedGlobalSectionsAb.{u, u + 1} B X hX).obj Q))

Tag 0F1H, a lemma

Tag 0F1H in the Stacks Project.

Stacks.EtaleCohomology.OpenImmersionProductSquare.isIso_baseChangeMap, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/OpenImmersionProductSquareBaseChange.lean, line 55.

Let K be a field, j : U ⟶ X an open immersion of schemes of finite type over K and Y a scheme of finite type over K. For every abelian sheaf 𝓕 on U_étale whose stalks are torsion of orders invertible in K, the base change map q⁻¹Rj_*𝓕 ⟶ Rh_*p⁻¹𝓕 of the square

Y ×_K X  ←-h-  Y ×_K U
   |q              |p
   X     ←-j-      U

is an isomorphism (in D⁺((Y ×_K X)_étale), 𝓕 placed in degree zero).

theorem isIso_baseChangeMap {K : Type u} [Field K] {X U Y : Scheme.{u}}
    (sX : X ⟶ Spec (.of K)) (sU : U ⟶ Spec (.of K)) (sY : Y ⟶ Spec (.of K))
    (j : U ⟶ X) [IsOpenImmersion j] (hj : j ≫ sX = sU) [LocallyOfFiniteType sX] [QuasiCompact sX]
    [LocallyOfFiniteType sU] [QuasiCompact sU] [LocallyOfFiniteType sY] [QuasiCompact sY]
    (𝓕 : Sheaf (smallEtaleTopology U) Ab.{u + 1})
    (h𝓕 : ∀ (t : GeometricPoint U) (a : (smallEtaleStalkAb t.hom).obj 𝓕),
      ∃ n : ℕ, 0 < n ∧ IsUnit ((n : ℤ) : K) ∧ n • a = 0) :
    IsIso (baseChangeMap sX sU sY j hj 𝓕)

Tag 0F1I, a lemma

Tag 0F1I in the Stacks Project.

Stacks.EtaleCohomology.BaseChangeOverField.isIso_smallEtaleDerivedBaseChangeAb_of_field, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/BaseChangeOverFieldTheorem.lean, line 248.

[Stacks, Tag 0F1I]. Let K be a field. For any commutative diagram

X  ←-p-  X'  ←-h-  Y
|        |f'       |e
Spec K ←-s- S' ←-g- T

of schemes over K with X' = X ×_{Spec K} S' and Y = X' ×_{S'} T and g quasi-compact and quasi-separated, and every abelian sheaf 𝓕 on T_étale whose stalks are torsion of orders invertible in K, the base change map (f')⁻¹Rg_*𝓕 ⟶ Rh_*e⁻¹𝓕 is an isomorphism (in D⁺(X'_étale), 𝓕 placed in degree zero).

𝓕 is the filtered colimit of its subsheaves 𝓕[n], n invertible in K (isIso_higherBaseChangeMapAb_of_forall_annihilated), and for those this is derivedBaseChangeHolds_of_isUnit.

theorem isIso_smallEtaleDerivedBaseChangeAb_of_field (K : Type u) [Field K] {X : Scheme.{u}}
    (πX : X ⟶ Spec (.of K)) {S' X' T Y : Scheme.{u}} (s : S' ⟶ Spec (.of K)) (p : X' ⟶ X)
    (f' : X' ⟶ S') (hp : IsPullback p f' πX s) (g : T ⟶ S') (h : Y ⟶ X') (e : Y ⟶ T)
    (sq : IsPullback h e f' g) [QuasiCompact g] [QuasiSeparated g]
    (𝓕 : Sheaf (smallEtaleTopology T) Ab.{u + 1})
    (h𝓕 : ∀ (t : GeometricPoint T) (a : (smallEtaleStalkAb t.hom).obj 𝓕),
      ∃ n : ℕ, 0 < n ∧ IsUnit ((n : ℤ) : K) ∧ n • a = 0) :
    IsIso ((smallEtaleDerivedBaseChangeAb f' g h e sq.w).app
      ((DerivedCategory.Plus.singleFunctor _ 0).obj 𝓕))

Tag 0F1J, a lemma

Tag 0F1J in the Stacks Project.

Stacks.EtaleCohomology.EtaleCoefficients.isIso_tensorBaseChangeMap_of_isUnit, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/TensorBaseChangeOverField.lean, line 84.

[Stacks, Tag 0F1J]. Let K be a field and n ≥ 1 an integer invertible in K. Consider a commutative diagram

X <-p-- X' <-h-- Y
|       |f'      |e
v       v        v
Spec K <- S' <-g- T

of schemes with X' = X ×_{Spec K} S' and Y = X' ×_{S'} T and g quasi-compact and quasi-separated. The canonical map

p⁻¹E ⊗^L_{ℤ/nℤ} (f')⁻¹Rg_*F ⟶ Rh_*(h⁻¹p⁻¹E ⊗^L_{ℤ/nℤ} e⁻¹F)

is an isomorphism if E ∈ D⁺(X_étale, ℤ/nℤ) has Tor amplitude in [a, ∞] for some a ∈ ℤ and F ∈ D⁺(T_étale, ℤ/nℤ).

This is isIso_tensorBaseChangeMap_zmod_of_derivedBaseChangeHolds, with derived base change over X and over the schemes finite over the affine opens of X given by [Stacks, Tag 0F1I] (derivedBaseChangeHolds_of_isUnit: X over K is arbitrary there). The lower bound on E (_hE, E ∈ D⁺) is part of the source's statement; the proof does not need it.

theorem isIso_tensorBaseChangeMap_of_isUnit (K : Type u) [Field K] (n : ℕ) [NeZero n]
    (hn : IsUnit (n : K)) {X X' Y S' T : Scheme.{u}} (πX : X ⟶ Spec (.of K))
    (πS : S' ⟶ Spec (.of K)) (p : X' ⟶ X) (f' : X' ⟶ S') (hX' : IsPullback p f' πX πS)
    (g : T ⟶ S') (h : Y ⟶ X') (e : Y ⟶ T) (hY : IsPullback h e f' g) [QuasiCompact g]
    [QuasiSeparated g] (E : DerivedModules (ULift.{u + 1} (ZMod n)) X) (_hE : ∃ b : ℤ, E.IsGE b)
    (a : ℤ) (hEa : HasTorAmplitudeGE (ULift.{u + 1} (ZMod n)) E a)
    (F : DerivedModules (ULift.{u + 1} (ZMod n)) T) (hF : ∃ b : ℤ, F.IsGE b) :
    IsIso (tensorBaseChangeMap (ULift.{u + 1} (ZMod n)) f' g h e hY.w
      ((pullback (ULift.{u + 1} (ZMod n)) p).obj E) F)

Tag 0F1N, a lemma

Tag 0F1N in the Stacks Project.

Stacks.EtaleCohomology.EtaleCoefficients.isIso_tensorBaseChangeMap_of_finiteType, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/TensorBaseChangeOverFieldFiniteType.lean, line 82.

[Stacks, Tag 0F1N]. Let K be a field and n ≥ 1 an integer invertible in K. Consider a commutative diagram

X <-p-- X' <-h-- Y
|       |f'      |e
v       v        v
Spec K <- S' <-g- T

of schemes of finite type over K (through πX, πS and g ≫ πS) with X' = X ×_K S' and Y = X' ×_{S'} T. Then the canonical map

p⁻¹E ⊗^L_{ℤ/nℤ} (f')⁻¹Rg_*F ⟶ Rh_*(h⁻¹p⁻¹E ⊗^L_{ℤ/nℤ} e⁻¹F)

is an isomorphism for all E ∈ D(X_étale, ℤ/nℤ) and F ∈ D(T_étale, ℤ/nℤ).

The reduction to E ∈ D⁺ of Tor amplitude bounded below and F ∈ D⁺, where the functors commute with direct sums because cd(g), cd(h) < ∞ ([Stacks, Tag 0F10]), is isIso_tensorBaseChangeMap_of_finiteType_of_isGE; the bounded below case is [Stacks, Tag 0F1J] (isIso_tensorBaseChangeMap_of_isUnit).

theorem isIso_tensorBaseChangeMap_of_finiteType (K : Type u) [Field K] (n : ℕ) [NeZero n]
    (hn : IsUnit (n : K)) {X X' Y S' T : Scheme.{u}} (πX : X ⟶ Spec (.of K))
    (πS : S' ⟶ Spec (.of K)) (p : X' ⟶ X) (f' : X' ⟶ S') (h : Y ⟶ X') (e : Y ⟶ T)
    (g : T ⟶ S') [LocallyOfFiniteType πX] [QuasiCompact πX] [LocallyOfFiniteType πS]
    [QuasiCompact πS] [LocallyOfFiniteType (g ≫ πS)] [QuasiCompact (g ≫ πS)]
    (hX' : IsPullback p f' πX πS) (hY : IsPullback h e f' g)
    (E : DerivedModules (ULift.{u + 1} (ZMod n)) X)
    (F : DerivedModules (ULift.{u + 1} (ZMod n)) T) :
    IsIso (tensorBaseChangeMap (ULift.{u + 1} (ZMod n)) f' g h e hY.w
      ((pullback (ULift.{u + 1} (ZMod n)) p).obj E) F)

Tag 0F1P, a lemma

Tag 0F1P in the Stacks Project.

Stacks.EtaleCohomology.EtaleCoefficients.isIso_kunnethMap_of_isSepClosed, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/KunnethOverSeparablyClosedFieldFiniteType.lean, line 101. References: SGA 4½ Th finitude (1.11), “special case”.

[Stacks, Tag 0F1P], as the Künneth map of [Stacks, Tag 0F1E]. Let k be a separably closed field, X and Y schemes of finite type over k (through f and g), and n ≥ 1 an integer invertible in k. For E ∈ D(X_étale, ℤ/nℤ) and K ∈ D(Y_étale, ℤ/nℤ) the Künneth map

RΓ(X, E) ⊗^L\_\{ℤ/nℤ} RΓ(Y, K) = Rf\_*E ⊗^L Rg\_*K ⟶ R(pr₁ ≫ f)\_*(pr₁⁻¹E ⊗^L pr₂⁻¹K) = RΓ(X ×\_k Y, pr₁⁻¹E ⊗^L pr₂⁻¹K)

is an isomorphism in D((Spec k)_étale, ℤ/nℤ) = D(ℤ/nℤ).

theorem isIso_kunnethMap_of_isSepClosed (hn : IsUnit (n : k))
    (E : DerivedModules (ULift.{u + 1} (ZMod n)) X)
    (K : DerivedModules (ULift.{u + 1} (ZMod n)) Y) :
    IsIso (kunnethMap (ULift.{u + 1} (ZMod n)) (Limits.pullback.fst f g) f
      (Limits.pullback.snd f g) g Limits.pullback.condition E K)

Stacks.EtaleCohomology.EtaleCoefficients.kunnethOverSepClosedIso, a def in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/KunnethOverSeparablyClosedFieldFiniteType.lean, line 120.

[Stacks, Tag 0F1P]. Let k be a separably closed field, X and Y schemes of finite type over k (through f and g), and n ≥ 1 an integer invertible in k. For E ∈ D(X_étale, ℤ/nℤ) and K ∈ D(Y_étale, ℤ/nℤ)

RΓ(X ×_k Y, pr₁⁻¹E ⊗^L_{ℤ/nℤ} pr₂⁻¹K) ≅ RΓ(X, E) ⊗^L_{ℤ/nℤ} RΓ(Y, K)

in D(ℤ/nℤ), where RΓ(Z, -) = Γ ∘ R(Z ⟶ Spec k)_* for the global sections equivalence Γ : D((Spec k)_étale, ℤ/nℤ) ≌ D(ℤ/nℤ) (derivedModulesEquivOfSepClosed). It is the inverse of Γ of the Künneth map of [Stacks, Tag 0F1E] (isIso_kunnethMap_of_isSepClosed), followed by the compatibility of Γ with ⊗^L (kunnethOverSepClosedIso_inv).

def kunnethOverSepClosedIso [HasDerivedCategory (ModuleCat.{u + 1} (ULift.{u + 1} (ZMod n)))]
    (hn : IsUnit (n : k))
    (E : DerivedModules (ULift.{u + 1} (ZMod n)) X)
    (K : DerivedModules (ULift.{u + 1} (ZMod n)) Y) :
    (derivedModulesEquivOfSepClosed k (ULift.{u + 1} (ZMod n))).functor.obj
        ((pushforward (ULift.{u + 1} (ZMod n)) (Limits.pullback.fst f g ≫ f)).obj
          (((tensor (ULift.{u + 1} (ZMod n)) (Limits.pullback f g)).obj
            ((pullback (ULift.{u + 1} (ZMod n)) (Limits.pullback.fst f g)).obj E)).obj
              ((pullback (ULift.{u + 1} (ZMod n)) (Limits.pullback.snd f g)).obj K))) ≅
      ((CochainComplex.DerivedTensor.canonicalDerivedTensorProduct
          (ULift.{u + 1} (ZMod n))).tensor.obj
        ((derivedModulesEquivOfSepClosed k (ULift.{u + 1} (ZMod n))).functor.obj
          ((pushforward (ULift.{u + 1} (ZMod n)) f).obj E))).obj
        ((derivedModulesEquivOfSepClosed k (ULift.{u + 1} (ZMod n))).functor.obj
          ((pushforward (ULift.{u + 1} (ZMod n)) g).obj K))

Stacks.EtaleCohomology.EtaleCoefficients.kunnethOverSepClosedIso_inv, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/EtaleCohomology/Kunneth/KunnethOverSeparablyClosedFieldFiniteType.lean, line 143. References: SGA 4½ Th finitude (1.11), “special case”.

[Stacks, Tag 0F1P]: the isomorphism is the Künneth map of [Stacks, Tag 0F1E]. The inverse of kunnethOverSepClosedIso is the compatibility of Γ with ⊗^L, RΓ(X, E) ⊗^L RΓ(Y, K) ≅ Γ(Rf_*E ⊗^L Rg_*K), followed by Γ of the Künneth map Rf_*E ⊗^L Rg_*K ⟶ R(pr₁ ≫ f)_*(pr₁⁻¹E ⊗^L pr₂⁻¹K) of Tag 0F1E.

theorem kunnethOverSepClosedIso_inv
    [HasDerivedCategory (ModuleCat.{u + 1} (ULift.{u + 1} (ZMod n)))] (hn : IsUnit (n : k))
    (E : DerivedModules (ULift.{u + 1} (ZMod n)) X)
    (K : DerivedModules (ULift.{u + 1} (ZMod n)) Y) :
    (kunnethOverSepClosedIso k n f g hn E K).inv =
      (((derivedModulesEquivOfSepClosedTensorIso k (ULift.{u + 1} (ZMod n))).app
          ((pushforward (ULift.{u + 1} (ZMod n)) f).obj E)).app
          ((pushforward (ULift.{u + 1} (ZMod n)) g).obj K)).inv ≫
        (derivedModulesEquivOfSepClosed k (ULift.{u + 1} (ZMod n))).functor.map
          (kunnethMap (ULift.{u + 1} (ZMod n)) (Limits.pullback.fst f g) f
            (Limits.pullback.snd f g) g Limits.pullback.condition E K)

Sources

• Section 59.97, Tag 0F13
• AxiomMath/Stacks at 5ed82d14

LeanStacks.Cohomology.KunnethMapOnRingedTopoi.directImageMap · Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfCupProduct · Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfAdjoint · Stacks.Cohomology.KunnethMapOnRingedTopoi.kunnethMapOfCupProduct_eq_kunnethMapOfAdjoint · Stacks.EtaleCohomology.isIso_etaleKunnethCompositeMap_zmod_of_finiteType · Stacks.EtaleCohomology.isIso_etaleKunnethCompositeMap_of_isGE_of_isProper · AlgebraicGeometry.subsingleton_sheafH_and_epi_counit_of_supportSheaf_subset_closedPoints · AlgebraicGeometry.isZero_iff_forall_isZero_etaleHypercohomology_of_closedPointSupport

Section59.98 Comparing chaotic and Zariski topologies#

Section 59.98 of the Stacks Project, Tag 0F1K. 1 tag here is carried by 1 declaration of the repository (in StacksFrontier), quoted below from AxiomMath/Stacks at commit 5ed82d14.

Tag 0F1L, a lemma

Tag 0F1L in the Stacks Project.

AlgebraicGeometry.SmallAffineZariski.essImage_iff_isZero_derivedSections_and_isCSquareIso, a theorem in StacksFrontier (drafted, not yet reviewed), declared in StacksFrontier/AlgebraicGeometry/Topologies/ZariskiTopology/ChaoticComparison.lean, line 207.

Stacks 0F1L. Let 𝒪 be a sheaf of rings for the Zariski topology on the affine opens of a scheme S and let ε be the comparison from the Zariski topology to the chaotic topology, in which every presheaf is a sheaf. A bounded-below object K of D(𝒪_chaotic) lies in the essential image of the derived direct image Rε_* if and only if the derived sections of K over every empty affine open vanish and the canonical map c^K_{U,V,W,V ∩ W} of Stacks 0F16 is an isomorphism for every square exhibiting an affine open U as the union of two standard opens V and W with intersection V ∩ W.

theorem essImage_iff_isZero_derivedSections_and_isCSquareIso
    (K : DerivedCategory (SheafOfModules.{u + 1}
      (Stacks.SheafOfModules.Tensor.ringSheaf (trivialRingedSite B S hS O).O)))
    (hK : (DerivedCategory.TStructure.t
      (C := SheafOfModules.{u + 1}
        (Stacks.SheafOfModules.Tensor.ringSheaf (trivialRingedSite B S hS O).O))).plus K) :
    Rε.essImage K ↔
      ((∀ (sq : Square (SmallAffineZariski B S hS)) (U : SmallAffineZariski B S hS),
          IsEmpty ((toScheme B S hS).obj U) →
            IsZero (((Stacks.MayerVietoris.CanonicalMapOfComplexes.derivedCSquareData.{u + 1,
              u + 1} (R := (Stacks.SheafOfModules.Tensor.ringSheaf (trivialRingedSite B S hS O).O))
              sq).derivedSections U).obj K)) ∧
        ∀ sq : Square (SmallAffineZariski B S hS), IsStandardOpenCoverSquare B S hS sq →
          (Stacks.MayerVietoris.CanonicalMapOfComplexes.derivedCSquareData.{u + 1, u + 1}
            (R := (Stacks.SheafOfModules.Tensor.ringSheaf (trivialRingedSite B S hS O).O))
            sq).IsCSquareIso K)

Sources

• Section 59.98, Tag 0F1K
• AxiomMath/Stacks at 5ed82d14

LeanAlgebraicGeometry.SmallAffineZariski.essImage_iff_isZero_derivedSections_and_isCSquareIso

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