File size: 4,302 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 | 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
end OAI
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