| import OAI.NumberTheory.PiExponent.Geometry.CoherentProjectiveFinite |
| import OAI.NumberTheory.PiExponent.Geometry.LineBundleInverse |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CoherentProjectiveFinite |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits |
| open PiExponentSeshadri.Geometry |
| variable {X Y : Scheme.{0}} |
|
|
| theorem positive_coherent_cohomology_finite_of_inverse_powers |
| [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))) |
| (hfinite : ∀ n q, 0 < q → |
| letI := Module.compHom (cohomology (L.pow n).inverse.sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (L.pow n).inverse.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) := |
| positive_coherent_cohomology_finite_of_standard_twists p L l hl s hcover haffine |
| (fun n => (L.pow n).inverse) (fun n => lineTensorInverseIso (L.pow n)) hfinite M q hq |
|
|
| theorem closed_positive_coherent_cohomology_finite_of_inverse_powers |
| [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))) |
| (hfinite : ∀ n q, 0 < q → |
| letI := Module.compHom (cohomology (L.pow n).inverse.sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (L.pow n).inverse.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) := |
| closed_positive_coherent_cohomology_finite_of_standard_twists f p L l hl s hcover haffine |
| (fun n => (L.pow n).inverse) (fun n => lineTensorInverseIso (L.pow n)) hfinite M q hq |
|
|
| theorem coherent_cohomology_finite_of_inverse_powers |
| [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))) |
| (hfinite : ∀ n q, |
| letI := Module.compHom (cohomology (L.pow n).inverse.sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (L.pow n).inverse.sheaf q)) |
| (M : Y.Modules) [M.IsFinitePresentation] (q : ℕ) : |
| letI := Module.compHom (cohomology M q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology M q) := |
| coherent_cohomology_finite_of_standard_twists p L l hl s hcover haffine |
| (fun n => (L.pow n).inverse) (fun n => lineTensorInverseIso (L.pow n)) hfinite M q |
|
|
| theorem closed_coherent_cohomology_finite_of_inverse_powers |
| [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))) |
| (hfinite : ∀ n q, |
| letI := Module.compHom (cohomology (L.pow n).inverse.sheaf q) (baseScalars p) |
| FiniteDimensional ℂ (cohomology (L.pow n).inverse.sheaf q)) |
| (M : X.Modules) [M.IsFinitePresentation] (q : ℕ) : |
| letI := Module.compHom (cohomology M q) (baseScalars (f ≫ p)) |
| FiniteDimensional ℂ (cohomology M q) := |
| closed_coherent_cohomology_finite_of_standard_twists f p L l hl s hcover haffine |
| (fun n => (L.pow n).inverse) (fun n => lineTensorInverseIso (L.pow n)) hfinite M q |
|
|
| end |
| end PiExponent.CoherentProjectiveFinite |
|
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| end OAI |
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|
