File size: 5,694 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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.NumberTheory.PiExponent.Ampleness.NakaiCartier
import OAI.NumberTheory.PiExponent.Approximation.FrameTensorPowers
namespace OAI
namespace PiExponent.CartierPowerFrames
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
variable {X : Scheme.{0}}
def sectionMultiplyLeft (L M : X.Modules) (s : structureSheaf X ⟶ L) :
M ⟶ moduleTensor X L M :=
(moduleTensorUnit M).inv ≫ moduleTensorMap s (𝟙 M)
def tensorFrame (L M : X.Modules) (U : X.Opens)
(e : L.restrict U.ι ≅ O U.toScheme)
(f : M.restrict U.ι ≅ O U.toScheme) :
(moduleTensor X L M).restrict U.ι ≅ O U.toScheme :=
moduleTensorRestrict U L M ≪≫ moduleTensorIso e f ≪≫
moduleTensorUnit (O U.toScheme)
def tensorUnitTwist (M : X.Modules) (U : X.Opens)
(f : M.restrict U.ι ≅ O U.toScheme) : O U.toScheme ≅ O U.toScheme :=
f.symm ≪≫ (Scheme.Modules.restrictFunctor U.ι).mapIso (moduleTensorUnit M).symm ≪≫
moduleTensorRestrict U (O X) M ≪≫
moduleTensorIso (Scheme.Modules.restrictUnitIso U.ι) f ≪≫ moduleTensorUnit (O U.toScheme)
private lemma frame_section {C : Type*} [Category C] {A B D : C}
(u : A ≅ B) (s : A ⟶ D) (e : D ≅ B) :
s ≫ e.hom = u.hom ≫ ((u.inv ≫ s) ≫ e.hom) := by
simp
private lemma framed_multiply {C : Type*} [Category C] {A B D P Q T O : C}
(g : O ⟶ A) (k : A ⟶ B) (m : B ⟶ D) (r : B ⟶ P) (r' : D ⟶ Q)
(e : P ⟶ T) (e' : Q ⟶ T) (u : T ⟶ O) (a : O ⟶ O)
(p : P ⟶ Q) (b : T ⟶ T)
(hn : m ≫ r' = r ≫ p) (ht : p ≫ e' = e ≫ b) (hs : b ≫ u = u ≫ a) :
g ≫ (k ≫ m) ≫ (r' ≫ e' ≫ u) = (g ≫ k ≫ r ≫ e ≫ u) ≫ a := by
simp only [Category.assoc]
rw [← Category.assoc m r', hn]
simp only [Category.assoc]
rw [← Category.assoc p e', ht]
simp only [Category.assoc]
rw [hs]
lemma tensor_multiply_framed (L M : X.Modules) (s : O X ⟶ L) (U : X.Opens)
(e : L.restrict U.ι ≅ O U.toScheme)
(f : M.restrict U.ι ≅ O U.toScheme) :
f.inv ≫ (Scheme.Modules.restrictFunctor U.ι).map (sectionMultiplyLeft L M s) ≫
(tensorFrame L M U e f).hom = (tensorUnitTwist M U f).hom ≫
(restrictSection U.ι s ≫ e.hom) := by
let F := Scheme.Modules.restrictFunctor U.ι
let R := moduleTensorRestrict U (O X) M
let R' := moduleTensorRestrict U L M
let E := moduleTensorIso (Scheme.Modules.restrictUnitIso U.ι) f
let E' := moduleTensorIso e f
let T := moduleTensorUnit (O U.toScheme)
let a : O U.toScheme ⟶ O U.toScheme := restrictSection U.ι s ≫ e.hom
let p := moduleTensorMap (F.map s) (𝟙 (M.restrict U.ι))
have hn : F.map (moduleTensorMap s (𝟙 M)) ≫ R'.hom = R.hom ≫ p :=
(moduleTensorRestrict_natural U s (𝟙 M)).trans
(congrArg (fun q => R.hom ≫ moduleTensorMap (F.map s) q) (F.map_id M))
have he : F.map s ≫ e.hom = (Scheme.Modules.restrictUnitIso U.ι).hom ≫ a := by
simpa only [a, restrictSection] using!
frame_section (Scheme.Modules.restrictUnitIso U.ι) (F.map s) e
have h₁ := (moduleTensorMap_comp (F.map s) e.hom (𝟙 (M.restrict U.ι)) f.hom).symm.trans
(congrArg (moduleTensorMap (F.map s ≫ e.hom)) (Category.id_comp f.hom))
have h₂ := (moduleTensorMap_comp (Scheme.Modules.restrictUnitIso U.ι).hom a
f.hom (𝟙 (O U.toScheme))).symm.trans
(congrArg (moduleTensorMap ((Scheme.Modules.restrictUnitIso U.ι).hom ≫ a))
(Category.comp_id f.hom))
have ht : p ≫ E'.hom = E.hom ≫ moduleTensorMap a (𝟙 (O U.toScheme)) :=
h₁.trans ((congrArg (fun q : F.obj (O X) ⟶ O U.toScheme => moduleTensorMap q f.hom)
he).trans h₂.symm)
have hs : moduleTensorMap a (𝟙 (O U.toScheme)) ≫ T.hom = T.hom ≫ a :=
(moduleTensorUnit_scalar (X := U.toScheme) a (𝟙 (O U.toScheme))).trans
(congrArg (fun q => T.hom ≫ q) (Category.comp_id a))
have hfinal := framed_multiply f.inv (F.map (moduleTensorUnit M).inv)
(F.map (moduleTensorMap s (𝟙 M))) R.hom R'.hom E.hom E'.hom T.hom a p _ hn ht hs
simpa only [sectionMultiplyLeft, tensorFrame, tensorUnitTwist] using!
(congrArg (fun q => f.inv ≫ q ≫ (tensorFrame L M U e f).hom)
(F.map_comp (moduleTensorUnit M).inv (moduleTensorMap s (𝟙 M)))).trans hfinal
lemma tensor_multiply_ideal (L M : X.Modules) (s : O X ⟶ L)
(U : X.affineOpens) (e : L.restrict U.1.ι ≅ O U.1.toScheme)
(f : M.restrict U.1.ι ≅ O U.1.toScheme) :
Ideal.span {U.1.topIso.hom (endValue (f.inv ≫
(Scheme.Modules.restrictFunctor U.1.ι).map (sectionMultiplyLeft L M s) ≫
(tensorFrame L M U.1 e f).hom))} =
Ideal.span {U.1.topIso.hom (coefficient e (restrictSection U.1.ι s))} := by
rw [tensor_multiply_framed,endValue_comp,map_mul,← Ideal.span_singleton_mul_span_singleton]
have hu : IsUnit (endValue (tensorUnitTwist M U.1 f).hom) :=
(end_isIso_iff _).mp inferInstance
rw [Ideal.span_singleton_eq_top.mpr (hu.map U.1.topIso.hom.hom),Ideal.top_mul]
rfl
theorem cartierPowerMultiply_ideal (L : LineBundle X) (s : GlobalSections X L.sheaf)
(n : ℕ) (U : X.affineOpens)
(e : L.sheaf.restrict U.1.ι ≅ structureSheaf U.1.toScheme) :
Ideal.span {U.1.topIso.hom (endValue ((modulePowFrame U.1 e n).inv ≫
(Scheme.Modules.restrictFunctor U.1.ι).map
(PiExponent.NumericalAmpleness.cartierPowerMultiply L s n) ≫
(modulePowFrame U.1 e (n+1)).hom))} =
Ideal.span {U.1.topIso.hom (coefficient e (restrictSection U.1.ι s))} :=
tensor_multiply_ideal L.sheaf (modulePow X L.sheaf n) s U e (modulePowFrame U.1 e n)
end
end PiExponent.CartierPowerFrames
end OAI
|