openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CurveFinitePushforwardCoherent.lean
| import OAI.NumberTheory.PiExponent.Approximation.ClosedPushforwardAffine | |
| import OAI.NumberTheory.PiExponent.Approximation.OpenBaseChange | |
| namespace OAI | |
| namespace PiExponent.CurveFinitePushforwardCoherent | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite | |
| open CoherentAffineFinite FiniteGlobalPresentation ClosedPushforwardAffine | |
| variable {X Y : Scheme.{0}} | |
| private def presentationOfIso (X : Scheme.{0}) | |
| {M N : SheafOfModules X.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_sections_finite [IsAffine X] [IsAffine Y] | |
| (f : X ⟶ Y) [IsFinite 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 | |
| let : Algebra Γ(Y,⊤) Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) := σ.toAlgebra | |
| let : Module.Finite Γ(Y,⊤) Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) := | |
| f.finite_app ⊤ (isAffineOpen_top Y) | |
| let : Module Γ(Y,⊤) Γ(M,f ⁻¹ᵁ (⊤ : Y.Opens)) := Module.compHom _ σ | |
| let : IsScalarTower Γ(Y,⊤) Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) Γ(M,f ⁻¹ᵁ (⊤ : Y.Opens)) := | |
| IsScalarTower.of_compHom _ _ _ | |
| have h := Module.Finite.trans (R := Γ(Y,⊤)) | |
| Γ(X,f ⁻¹ᵁ (⊤ : Y.Opens)) Γ(M,f ⁻¹ᵁ (⊤ : Y.Opens)) | |
| exact h | |
| theorem affine_pushforward_exists_finitePresentation [IsAffine X] [IsAffine Y] | |
| [IsLocallyNoetherian Y] (f : X ⟶ Y) [IsFinite 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) | |
| instance pushforward_isFinitePresentation (f : X ⟶ Y) [IsFinite f] | |
| [IsLocallyNoetherian Y] (M : X.Modules) [M.IsFinitePresentation] : | |
| ((Scheme.Modules.pushforward f).obj M).IsFinitePresentation := by | |
| let : M.IsQuasicoherent := | |
| (SheafOfModules.IsFinitePresentation.exists_quasicoherentData M).choose.isQuasicoherent | |
| apply isFinitePresentation_of_affine_presentations | |
| intro U | |
| have : IsAffine U.1.toScheme := U.2 | |
| have : IsAffine (f ⁻¹ᵁ U.1).toScheme := U.2.preimage f | |
| have hM := (locallyFinitelyGenerated_of_finitePresentation M).restrict (f ⁻¹ᵁ U.1).ι | |
| let : (M.restrict (f ⁻¹ᵁ U.1).ι).IsQuasicoherent := | |
| Scheme.Modules.isQuasicoherent_restrictFunctor (f ⁻¹ᵁ U.1).ι M | |
| obtain ⟨P,hP⟩ := affine_pushforward_exists_finitePresentation (f ∣_ U.1) | |
| (M.restrict (f ⁻¹ᵁ U.1).ι) hM | |
| exact ⟨presentationOfIso U.1.toScheme | |
| (PiExponentSeshadri.OpenBaseChange.iso f U.1 M).symm P, | |
| ⟨⟨hP.isFiniteType_generators.finite⟩, ⟨hP.isFiniteType_relations.finite⟩⟩⟩ | |
| end | |
| end PiExponent.CurveFinitePushforwardCoherent | |
| end OAI | |