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a72760f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 | import OAI.NumberTheory.PiExponent.Ampleness.AmpleAllPowersGeneration
import OAI.NumberTheory.PiExponent.Approximation.SectionOpens
import OAI.NumberTheory.PiExponent.Cohomology.FiniteCoverCohomology
namespace OAI
namespace PiExponent.GeneratorsSectionCover
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames PiExponentSeshadri.SectionOpens
variable {X : Scheme.{0}}
private lemma epi_iso_comp {C : Type*} [Category C] {A B D E : C}
(i : A ≅ B) (f : B ⟶ D) (e : D ≅ E) [Epi f] :
Epi (i.hom ≫ f ≫ e.hom) := inferInstance
private abbrev schemeFree (X : Scheme.{0}) (ι : Type) : X.Modules :=
SheafOfModules.free ι
private abbrev schemeInclusion {X : Scheme.{0}} {ι : Type} (i : ι) :
O X ⟶ schemeFree X ι := SheafOfModules.ιFree i
def freeProjection {ι : Type} [DecidableEq ι] (i : ι) :
schemeFree X ι ⟶ O X :=
Sigma.desc (C := X.Modules) (f := fun _ : ι => O X) (fun j => if j = i then 𝟙 _ else 0)
@[simp] theorem free_inclusion_projection {ι : Type} [DecidableEq ι] (i j : ι) :
schemeInclusion j ≫ freeProjection (X := X) i = if j = i then 𝟙 _ else 0 := by
exact Sigma.ι_comp_desc (C := X.Modules) (f := fun _ : ι => O X) _ _
theorem free_sum_projection_inclusion {ι : Type} [Fintype ι] [DecidableEq ι] :
(∑ i : ι, freeProjection (X := X) i ≫ schemeInclusion i) =
𝟙 (schemeFree X ι) := by
apply (SheafOfModules.isColimitFreeCofan ι).hom_ext
intro j
change schemeInclusion j.as ≫ _ = schemeInclusion j.as ≫ _
simp only [Preadditive.comp_sum, ← Category.assoc, free_inclusion_projection]
simp only [ite_comp, Category.id_comp, zero_comp, Fintype.sum_ite_eq, Category.comp_id]
lemma hom_sum_app {M N : X.Modules} {ι : Type} (S : Finset ι)
(f : ι → (M ⟶ N)) (V : X.Opens) (z : Γ(M,V)) :
(∑ i ∈ S, f i).app V z = ∑ i ∈ S, (f i).app V z := by
classical
induction S using Finset.induction_on with
| empty => rfl
| @insert a S ha ih =>
simp only [Finset.sum_insert ha, Scheme.Modules.Hom.add_app]
change (f a).app V z + (∑ i ∈ S, f i).app V z = _
rw [ih]
theorem unit_epimorphism_cover {ι : Type} [Finite ι]
(f : schemeFree X ι ⟶ O X) [Epi f] :
(⨆ i : ι, isoOpen (schemeInclusion i ≫ f)) = ⊤ := by
classical
let : Fintype ι := Fintype.ofFinite ι
apply top_unique
intro x _
let F : X.Modules ⥤ _ := SheafOfModules.toSheaf X.ringCatSheaf
have : F.PreservesEpimorphisms :=
inferInstanceAs (SheafOfModules.toSheaf X.ringCatSheaf).PreservesEpimorphisms
have hFf : Epi (F.map f) := F.map_epi f
have hloc : TopCat.Presheaf.IsLocallySurjective (F.map f).hom :=
(TopCat.Sheaf.isLocallySurjective_iff_epi _).mpr hFf
obtain ⟨V, hV, ⟨z, hz⟩, hxV⟩ :=
(TopCat.Presheaf.isLocallySurjective_iff _).mp hloc ⊤ (1 : Γ(X,⊤)) x trivial
change Scheme.Modules.Hom.app f V z = X.presheaf.map (homOfLE hV).op (1 : Γ(X,⊤)) at hz
rw [map_one] at hz
have he : (∑ i : ι, freeProjection (X := X) i ≫ (schemeInclusion i ≫ f)) = f := by
simp only [← Category.assoc]
rw [← Preadditive.sum_comp, free_sum_projection_inclusion, Category.id_comp]
have hsum : (∑ i : ι, Scheme.Modules.Hom.app (schemeInclusion i ≫ f) V
(Scheme.Modules.Hom.app (freeProjection (X := X) i) V z)) = (1 : Γ(X,V)) := by
exact (hom_sum_app Finset.univ
(fun i => freeProjection (X := X) i ≫ (schemeInclusion i ≫ f)) V z).symm.trans
((congrArg (fun q : (schemeFree X ι : X.Modules) ⟶ O X =>
Scheme.Modules.Hom.app q V z) he).trans hz)
have hsum' : (∑ i : ι, X.presheaf.germ V x hxV
(Scheme.Modules.Hom.app (schemeInclusion i ≫ f) V (Scheme.Modules.Hom.app (freeProjection (X := X) i) V z))) = 1 := by
exact (map_sum (X.presheaf.germ V x hxV).hom _ _).symm.trans
((congrArg (X.presheaf.germ V x hxV) hsum).trans (X.presheaf.germ V x hxV).hom.map_one)
obtain ⟨i, _, hi⟩ := IsLocalRing.exists_of_isUnit_sum (hsum' ▸ isUnit_one)
have hc : IsUnit (X.presheaf.germ ⊤ x trivial
(endValue (schemeInclusion i ≫ f))) := by
let si : O X ⟶ O X := schemeInclusion i ≫ f
let a : Γ(X,V) := Scheme.Modules.Hom.app (freeProjection (X := X) i) V z
have ht := end_apply si V a
have hi' : IsUnit (X.presheaf.germ V x hxV
(a * (show Γ(X,V) from si.app V (1 : Γ(X,V))))) := ht ▸ hi
have hc' : IsUnit (X.presheaf.germ V x hxV
(show Γ(X,V) from si.app V (1 : Γ(X,V)))) := by
rw [map_mul] at hi'
exact isUnit_of_mul_isUnit_right hi'
have hcoeff := congrArg (fun z : Γ(X,V) => X.presheaf.germ V x hxV z)
(end_naturality si V)
rw [X.presheaf.germ_res_apply] at hcoeff
exact hcoeff ▸ hc'
apply Opens.mem_iSup.mpr
refine ⟨i, ?_⟩
erw [isoOpen_eq_basicOpen (Iso.refl (O X))]
change x ∈ X.basicOpen (endValue (schemeInclusion i ≫ f))
exact (X.mem_basicOpen (endValue (schemeInclusion i ≫ f)) x (show x ∈ (⊤ : X.Opens) from trivial)).mpr hc
def generatorSection (L : LineBundle X) (G : L.sheaf.GeneratingSections) (i : G.I) :
GlobalSections X L.sheaf := schemeInclusion i ≫ G.π
theorem generators_sectionOpen_cover (L : LineBundle X)
(G : L.sheaf.GeneratingSections) [G.IsFiniteType] :
(⨆ i : G.I, sectionOpen X (generatorSection L G i)) = ⊤ := by
apply top_unique
intro x _
obtain ⟨U, hx, ⟨e⟩⟩ := L.locallyRankOne x
let F : SheafOfModules X.ringCatSheaf ⥤ SheafOfModules U.toScheme.ringCatSheaf :=
Scheme.Modules.restrictFunctor U.ι
have : PreservesColimitsOfSize.{0,0} F :=
inferInstanceAs (PreservesColimitsOfSize.{0,0} (Scheme.Modules.restrictFunctor U.ι))
let η := (Scheme.Modules.restrictUnitIso U.ι).symm
let f : schemeFree U.toScheme G.I ⟶ O U.toScheme :=
(SheafOfModules.mapFreeIso F G.I η).hom ≫ F.map G.π ≫ e.hom
have hπ : Epi (F.map G.π) := @Functor.map_epi _ _ _ _ F _ _ _ G.π G.epi
have : Epi f := @epi_iso_comp _ _ _ _ _ _
(SheafOfModules.mapFreeIso F G.I η) (F.map G.π) e hπ
let : Finite G.I := (inferInstance : G.IsFiniteType).finite
have hf := unit_epimorphism_cover f
obtain ⟨i, hi⟩ := Opens.mem_iSup.mp
(show (⟨x,hx⟩ : U.toScheme) ∈ ⨆ i : G.I, isoOpen (schemeInclusion i ≫ f) by
rw [hf]; trivial)
have he : schemeInclusion i ≫ f =
restrictSection U.ι (generatorSection L G i) ≫ e.hom := by
dsimp only [f, generatorSection, restrictSection]
erw [← Category.assoc, ← Category.assoc, SheafOfModules.ιFree_mapFreeIso_hom]
erw [F.map_comp]
exact Category.assoc _ _ _
erw [he, isoOpen_postcomp, isoOpen_restrictSection] at hi
exact Opens.mem_iSup.mpr ⟨i, hi⟩
theorem exists_fin_section_cover (L : LineBundle X)
(G : L.sheaf.GeneratingSections) [G.IsFiniteType] :
∃ k : ℕ, ∃ s : Fin k → GlobalSections X L.sheaf,
(⨆ i, sectionOpen X (s i)) = ⊤ := by
let : Finite G.I := (inferInstance : G.IsFiniteType).finite
let : Fintype G.I := Fintype.ofFinite G.I
let e := (Fintype.equivFin G.I).symm
exact ⟨Fintype.card G.I, fun i => generatorSection L G (e i),
(e.iSup_comp (g := fun i => sectionOpen X (generatorSection L G i))).trans
(generators_sectionOpen_cover L G)⟩
theorem ample_eventual_fin_section_cover [NoetherianSpace X]
(L : LineBundle X) (hL : L.IsAmple) :
∃ N : ℕ, ∀ n ≥ N, ∃ k : ℕ,
∃ s : Fin k → GlobalSections X (modulePow X L.sheaf n),
(⨆ i, sectionOpen X (s i)) = ⊤ := by
obtain ⟨N, hN⟩ := AmpleGlobalGeneration.ample_eventual_power_global_generators L hL
refine ⟨N, fun n hn => ?_⟩
obtain ⟨G, hG⟩ := hN n hn
let : G.IsFiniteType := hG
exact @exists_fin_section_cover X (L.pow n) G hG
end
end PiExponent.GeneratorsSectionCover
end OAI
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