File size: 4,676 Bytes
a72760f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import OAI.NumberTheory.PiExponent.Approximation.GlobalSectionClearing
import OAI.NumberTheory.PiExponent.LocalAlgebra.FiniteGlobalPresentation
namespace OAI
namespace PiExponent.ClosedPushforwardAffine
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite
open CoherentAffineFinite FiniteGlobalPresentation
variable {X Y : Scheme.{0}}
private def transportPresentation {Z : Scheme.{0}}
{M N : SheafOfModules Z.ringCatSheaf} (e : M ≅ N) (P : M.Presentation) :
N.Presentation := by
let : IsIso e.hom := e.isIso_hom
exact P.ofIsIso e.hom
theorem affine_pushforward_quasicoherent [IsAffine X] [IsAffine Y]
(f : X ⟶ Y) (M : X.Modules) [M.IsQuasicoherent] :
((Scheme.Modules.pushforward f).obj M).IsQuasicoherent := by
let MX := (Scheme.Modules.pushforward X.isoSpec.hom).obj M
have : MX.IsQuasicoherent := GlobalSectionClearing.quasicoherent_pushforward_iso X.isoSpec M
let MY := (Scheme.Modules.pushforward (Spec.map f.appTop)).obj MX
have : MY.IsQuasicoherent :=
(isQuasicoherent_iff_isIso_fromTildeΓ MY).mpr (isIso_fromTildeΓ_pushforward f.appTop MX)
have hback : ((Scheme.Modules.pushforward Y.isoSpec.inv).obj MY).IsQuasicoherent :=
GlobalSectionClearing.quasicoherent_pushforward_iso Y.isoSpec.symm MY
let e₁ := (Scheme.Modules.pushforward Y.isoSpec.inv).mapIso
((Scheme.Modules.pushforwardComp X.isoSpec.hom (Spec.map f.appTop)).app M)
let e₂ := (Scheme.Modules.pushforwardComp (X.isoSpec.hom ≫ Spec.map f.appTop) Y.isoSpec.inv).app M
have hf : (X.isoSpec.hom ≫ Spec.map f.appTop) ≫ Y.isoSpec.inv = f := by
rw [Scheme.isoSpec_hom_naturality, Category.assoc, Iso.hom_inv_id, Category.comp_id]
let e₃ := (Scheme.Modules.pushforwardCongr hf).app M
exact (SheafOfModules.isQuasicoherent Y.ringCatSheaf).prop_of_iso (e₁ ≪≫ e₂ ≪≫ e₃) hback
theorem affine_pushforward_sections_finite [IsAffine X] [IsAffine Y]
(f : X ⟶ Y) [IsClosedImmersion f] (M : X.Modules) [M.IsQuasicoherent]
(hM : LocallyFinitelyGenerated M) :
Module.Finite Γ(Y,⊤) Γ((Scheme.Modules.pushforward f).obj M,⊤) := by
have hfin : Module.Finite Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) Γ(M,f ⁻¹ᵁ (⊤ : Y.Opens)) := by
simpa only [Scheme.Hom.preimage_top] using! affine_sections_finite_of_localGenerators M hM
let σ : Γ(Y,⊤) →+* Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) := (f.app ⊤).hom
have : RingHomSurjective σ := ⟨f.app_surjective ⊤ (isAffineOpen_top Y)⟩
let ψ : Γ((Scheme.Modules.pushforward f).obj M,⊤) →ₛₗ[σ] Γ(M,f ⁻¹ᵁ (⊤ : Y.Opens)) := {
toFun := fun x => x
map_add' _ _ := rfl
map_smul' _ _ := rfl }
exact (ψ.finite_iff_of_bijective ⟨fun _ _ h => h, fun x => ⟨x, rfl⟩⟩).mpr hfin
theorem affine_exists_finitePresentation_of_sections [IsAffine X] [IsLocallyNoetherian X]
(M : X.Modules) [M.IsQuasicoherent] [Module.Finite Γ(X,⊤) Γ(M,⊤)] :
∃ P : M.Presentation, P.IsFinite := by
let a := X.isoSpec.inv
let b := X.isoSpec.hom
have : IsNoetherianRing Γ(X,⊤) :=
IsLocallyNoetherian.component_noetherian ⟨⊤, isAffineOpen_top X⟩
have : Module.Finite Γ(X,⊤) ((modulesSpecToSheaf.obj (M.restrict a)).obj.obj (op ⊤)) :=
finite_isoSpec_sections M
obtain ⟨P,hP⟩ := spec_exists_finitePresentation_of_sections (M.restrict a)
let F : SheafOfModules (Spec Γ(X,⊤)).ringCatSheaf ⥤ SheafOfModules X.ringCatSheaf :=
Scheme.Modules.restrictFunctor b
let : PreservesColimitsOfSize.{0,0} F :=
(Scheme.Modules.restrictAdjunction b).leftAdjoint_preservesColimits
let Q := P.map F (Scheme.Modules.restrictUnitIso b).symm
let e₁ := (Scheme.Modules.restrictFunctorComp b a).app M
have e₂ : M.restrict (𝟙 X) ≅ ((M.restrict a).restrict b) := by
change M.restrict (b ≫ a) ≅ ((M.restrict a).restrict b) at e₁
simpa only [a, b, Iso.hom_inv_id] using e₁
let e₃ := e₂.symm ≪≫ Scheme.Modules.restrictFunctorId.app M
exact ⟨transportPresentation (Z := X) e₃ Q,
⟨⟨hP.isFiniteType_generators.finite⟩, ⟨hP.isFiniteType_relations.finite⟩⟩⟩
theorem affine_pushforward_exists_finitePresentation [IsAffine X] [IsAffine Y]
[IsLocallyNoetherian Y] (f : X ⟶ Y) [IsClosedImmersion f]
(M : X.Modules) [M.IsQuasicoherent] (hM : LocallyFinitelyGenerated M) :
∃ P : ((Scheme.Modules.pushforward f).obj M).Presentation, P.IsFinite := by
have := affine_pushforward_quasicoherent f M
have := affine_pushforward_sections_finite f M hM
exact affine_exists_finitePresentation_of_sections ((Scheme.Modules.pushforward f).obj M)
end
end PiExponent.ClosedPushforwardAffine
end OAI
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