| import OAI.NumberTheory.PiExponent.Ampleness.AmpleGlobalGeneration |
| import OAI.NumberTheory.PiExponent.Approximation.ClosedPushforwardLinear |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PositiveCoherentExtFiniteness |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CoherentProjectiveFinite |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits |
| open PiExponentSeshadri.Geometry |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| theorem coherent_cohomology_finite_of_standard_twists |
| [IsNoetherian Y] [IsAffineHom (pullback.diagonal (terminal.from Y))] |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) |
| (L : LineBundle Y) (l : ℕ) (hl : 0 < l) |
| (s : Fin l → (structureSheaf Y ⟶ L.sheaf)) |
| (hcover : (⨆ i, PiExponentSeshadri.SectionOpens.isoOpen (s i)) = ⊤) |
| (haffine : ∀ i, IsAffineOpen (PiExponentSeshadri.SectionOpens.isoOpen (s i))) |
| (N : ℕ → LineBundle Y) |
| (hpair : ∀ n, moduleTensor Y (L.pow n).sheaf (N n).sheaf ≅ structureSheaf Y) |
| (hfinite : ∀ n q, letI := Module.compHom (cohomology (N n).sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (N n).sheaf q)) |
| (M : Y.Modules) [M.IsFinitePresentation] (q : ℕ) : |
| letI := Module.compHom (cohomology M q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology M q) := by |
| apply CoherentExtFiniteness.coherent_cohomology_finite_of_presentations p |
| l hl (fun i => PiExponentSeshadri.SectionOpens.isoOpen (s i)) haffine hcover _ M q |
| intro A hA |
| obtain ⟨n, hn⟩ := AmpleGlobalGeneration.eventual_global_generators_of_section_cover |
| L A s hcover haffine |
| obtain ⟨g, hg⟩ := hn n le_rfl |
| let : g.IsFiniteType := hg |
| exact ⟨NegativeTwistPresentations.coherentFinitePresentation L (N n) g (hpair n) p |
| (hfinite n)⟩ |
|
|
| theorem closed_coherent_cohomology_finite_of_standard_twists |
| [IsNoetherian Y] [IsAffineHom (pullback.diagonal (terminal.from Y))] |
| (f : X ⟶ Y) [IsClosedImmersion f] |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) |
| (L : LineBundle Y) (l : ℕ) (hl : 0 < l) |
| (s : Fin l → (structureSheaf Y ⟶ L.sheaf)) |
| (hcover : (⨆ i, PiExponentSeshadri.SectionOpens.isoOpen (s i)) = ⊤) |
| (haffine : ∀ i, IsAffineOpen (PiExponentSeshadri.SectionOpens.isoOpen (s i))) |
| (N : ℕ → LineBundle Y) |
| (hpair : ∀ n, moduleTensor Y (L.pow n).sheaf (N n).sheaf ≅ structureSheaf Y) |
| (hfinite : ∀ n q, letI := Module.compHom (cohomology (N n).sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (N n).sheaf q)) |
| (M : X.Modules) [M.IsFinitePresentation] (q : ℕ) : |
| letI := Module.compHom (cohomology M q) (baseScalars (f ≫ p)) |
| FiniteDimensional ℂ (cohomology M q) := by |
| exact ClosedImmersionSerreTransfer.finiteDimensional_of_pushforward f p M q |
| (coherent_cohomology_finite_of_standard_twists p L l hl s hcover haffine N hpair hfinite |
| ((Scheme.Modules.pushforward f).obj M) q) |
|
|
| theorem positive_coherent_cohomology_finite_of_standard_twists |
| [IsNoetherian Y] [IsAffineHom (pullback.diagonal (terminal.from Y))] |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) |
| (L : LineBundle Y) (l : ℕ) (hl : 0 < l) |
| (s : Fin l → (structureSheaf Y ⟶ L.sheaf)) |
| (hcover : (⨆ i, PiExponentSeshadri.SectionOpens.isoOpen (s i)) = ⊤) |
| (haffine : ∀ i, IsAffineOpen (PiExponentSeshadri.SectionOpens.isoOpen (s i))) |
| (N : ℕ → LineBundle Y) |
| (hpair : ∀ n, moduleTensor Y (L.pow n).sheaf (N n).sheaf ≅ structureSheaf Y) |
| (hfinite : ∀ n q, 0 < q → letI := Module.compHom (cohomology (N n).sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (N n).sheaf q)) |
| (M : Y.Modules) [M.IsFinitePresentation] (q : ℕ) (hq : 0 < q) : |
| letI := Module.compHom (cohomology M q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology M q) := by |
| apply CoherentExtFiniteness.positive_coherent_cohomology_finite_of_presentations p |
| l hl (fun i => PiExponentSeshadri.SectionOpens.isoOpen (s i)) haffine hcover _ M q hq |
| intro A hA |
| obtain ⟨n, hn⟩ := AmpleGlobalGeneration.eventual_global_generators_of_section_cover |
| L A s hcover haffine |
| obtain ⟨g, hg⟩ := hn n le_rfl |
| let : g.IsFiniteType := hg |
| exact ⟨NegativeTwistPresentations.positiveCoherentFinitePresentation L (N n) g (hpair n) p |
| (hfinite n)⟩ |
|
|
| theorem closed_positive_coherent_cohomology_finite_of_standard_twists |
| [IsNoetherian Y] [IsAffineHom (pullback.diagonal (terminal.from Y))] |
| (f : X ⟶ Y) [IsClosedImmersion f] |
| (p : Y ⟶ Spec (CommRingCat.of ℂ)) |
| (L : LineBundle Y) (l : ℕ) (hl : 0 < l) |
| (s : Fin l → (structureSheaf Y ⟶ L.sheaf)) |
| (hcover : (⨆ i, PiExponentSeshadri.SectionOpens.isoOpen (s i)) = ⊤) |
| (haffine : ∀ i, IsAffineOpen (PiExponentSeshadri.SectionOpens.isoOpen (s i))) |
| (N : ℕ → LineBundle Y) |
| (hpair : ∀ n, moduleTensor Y (L.pow n).sheaf (N n).sheaf ≅ structureSheaf Y) |
| (hfinite : ∀ n q, 0 < q → letI := Module.compHom (cohomology (N n).sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (N n).sheaf q)) |
| (M : X.Modules) [M.IsFinitePresentation] (q : ℕ) (hq : 0 < q) : |
| letI := Module.compHom (cohomology M q) (baseScalars (f ≫ p)) |
| FiniteDimensional ℂ (cohomology M q) := by |
| exact ClosedImmersionSerreTransfer.finiteDimensional_of_pushforward f p M q |
| (positive_coherent_cohomology_finite_of_standard_twists p L l hl s hcover haffine N hpair hfinite |
| ((Scheme.Modules.pushforward f).obj M) q hq) |
|
|
| end |
| end PiExponent.CoherentProjectiveFinite |
|
|
| end OAI |
|
|
