openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedPushforwardAffine.lean
| 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 | |