File size: 3,247 Bytes
b797a5b | 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 | 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
|