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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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 | 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
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