File size: 6,608 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.NumberTheory.PiExponent.Approximation.SectionOpens
import OAI.NumberTheory.PiExponent.Geometry.SupportedSheaf
namespace OAI
namespace PiExponentSeshadri.Geometry
noncomputable section
open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite
open PiExponentSeshadri.Frames PiExponentSeshadri.SectionOpens
variable {X : Scheme.{0}}
def openSectionModule (M : X.Modules) (U : X.Opens) : ModuleCat Γ(X,⊤) := by
letI := Module.compHom Γ(M,U) (X.presheaf.map (homOfLE (show U ≤ ⊤ from le_top)).op).hom
exact ModuleCat.of Γ(X,⊤) Γ(M,U)
def openSectionRestriction (M : X.Modules) (U : X.Opens) :
Γ(M,⊤) →ₗ[Γ(X,⊤)] openSectionModule M U where
toFun := M.presheaf.map (homOfLE (show U ≤ ⊤ from le_top)).op
map_add' := map_add _
map_smul' := fun r s => M.map_smul _ r s
def disjointSupportCoverEquiv (M : X.Modules) {ι : Type*} (D : ι → X.Opens)
(hcover : ⊤ ≤ ⨆ i, D i)
(hz : ∀ i j, i ≠ j → Subsingleton Γ(M,D i ⊓ D j)) :
Γ(M,⊤) ≃ₗ[Γ(X,⊤)] ((i : ι) → openSectionModule M (D i)) := by
let F : (X : TopCat).Sheaf AddCommGrpCat :=
(SheafOfModules.toSheaf X.ringCatSheaf).obj M
let f := LinearMap.pi (fun i => openSectionRestriction M (D i))
apply LinearEquiv.ofBijective f
constructor
· intro s t h
apply F.eq_of_locally_eq' D ⊤ (fun _ => homOfLE le_top) hcover s t
intro i
exact congrFun h i
· intro s
have hc : TopCat.Presheaf.IsCompatible F.obj D s := by
intro i j
by_cases h : i = j
· subst j; rfl
· let : Subsingleton (F.obj.obj (op (D i ⊓ D j))) := hz i j h
exact Subsingleton.elim _ _
obtain ⟨t,ht,_⟩ := F.existsUnique_gluing' D ⊤ (fun _ => homOfLE le_top) hcover s hc
exact ⟨t,funext ht⟩
def openSectionIso (M N : X.Modules) (U : X.Opens)
(e : M.restrict U.ι ≅ N.restrict U.ι) :
openSectionModule M U ≃ₗ[Γ(X,⊤)] openSectionModule N U := by
let etop := ((Scheme.Modules.toPresheaf U.toScheme).mapIso e).app (op ⊤)
let a := (M.restrictAppIso U.ι ⊤).symm ≪≫ etop ≪≫ N.restrictAppIso U.ι ⊤
have E : openSectionModule M (U.ι ''ᵁ ⊤) ≃ₗ[Γ(X,⊤)]
openSectionModule N (U.ι ''ᵁ ⊤) := by
refine { toFun := fun x => a.hom x
invFun := fun x => a.inv x
left_inv := by intro s; exact ConcreteCategory.congr_hom a.hom_inv_id s
right_inv := by intro s; exact ConcreteCategory.congr_hom a.inv_hom_id s
map_add' := map_add _
map_smul' := ?_ }
intro r s
change Γ(M,U.ι ''ᵁ ⊤) at s
change (N.restrictAppIso U.ι ⊤).hom
(e.hom.app ⊤ ((M.restrictAppIso U.ι ⊤).inv
((X.presheaf.map (homOfLE (show U.ι ''ᵁ ⊤ ≤ ⊤ from le_top)).op r) • s))) =
(X.presheaf.map (homOfLE (show U.ι ''ᵁ ⊤ ≤ ⊤ from le_top)).op r) • a.hom s
rw [Scheme.Modules.smul_restrictAppIso_inv_apply,e.hom.app_smul,
Scheme.Modules.smul_restrictAppIso_hom_apply]
simp only [← ConcreteCategory.comp_apply, Category.assoc, Iso.hom_inv_id, Category.comp_id]
rfl
have htop : U.ι ''ᵁ (⊤ : U.toScheme.Opens) = U := by
rw [Scheme.Hom.image_top_eq_opensRange,Scheme.Opens.opensRange_ι]
exact htop ▸ E
def finiteSupportLocallyIsoGlobal (M N : X.Modules) (S : Set X)
(hfin : S.Finite) (hclosed : ∀ x ∈ S, IsClosed ({x} : Set X))
(hM : ∀ x ∉ S, Subsingleton (M.presheaf.stalk x))
(hN : ∀ x ∉ S, Subsingleton (N.presheaf.stalk x))
(hloc : ∀ x ∈ S, ∃ U : X.Opens, x ∈ U ∧ Nonempty (M.restrict U.ι ≅ N.restrict U.ι)) :
Γ(M,⊤) ≃ₗ[Γ(X,⊤)] Γ(N,⊤) := by
classical
choose U hx he using fun x : S => hloc x x.property
let e : ∀ x : S, M.restrict (U x).ι ≅ N.restrict (U x).ι := fun x => (he x).some
have hc : IsClosed S := by
rw [← Set.biUnion_of_singleton S]
exact hfin.isClosed_biUnion hclosed
have hcx (x : S) : IsClosed (S \ {x.val}) := by
rw [← Set.biUnion_of_singleton (S \ {x.val})]
exact (hfin.subset Set.sdiff_subset).isClosed_biUnion (fun y hy => hclosed y hy.1)
let V : S → X.Opens := fun x => U x ⊓ ⟨(S \ {x.val})ᶜ,(hcx x).isOpen_compl⟩
let D : Option S → X.Opens := fun x => match x with
| none => ⟨Sᶜ,hc.isOpen_compl⟩
| some x => V x
have hV (x : S) : x.val ∈ V x := ⟨hx x,fun h => h.2 rfl⟩
have hcover : ⊤ ≤ ⨆ i, D i := by
intro x _
by_cases hs : x ∈ S
· exact Opens.mem_iSup.mpr ⟨some ⟨x,hs⟩,hV ⟨x,hs⟩⟩
· exact Opens.mem_iSup.mpr ⟨none,hs⟩
have honly (x : X) (hs : x ∈ S) (i : Option S) (hi : x ∈ D i) :
i = some ⟨x,hs⟩ := by
cases i with
| none => exact (hi hs).elim
| some y =>
congr 1
apply Subtype.ext
by_contra hne
exact hi.2 ⟨hs,fun h => hne h.symm⟩
have hz (P : X.Modules) (hP : ∀ x ∉ S, Subsingleton (P.presheaf.stalk x)) :
∀ i j, i ≠ j → Subsingleton Γ(P,D i ⊓ D j) := by
intro i j hij
apply FiniteSupport.sections_subsingleton_of_stalks
((SheafOfModules.toSheaf X.ringCatSheaf).obj P)
intro x hx
apply hP
intro hs
exact hij ((honly x hs i hx.1).trans (honly x hs j hx.2).symm)
have eD (i : Option S) : openSectionModule M (D i) ≃ₗ[Γ(X,⊤)] openSectionModule N (D i) := by
cases i with
| none =>
letI : Subsingleton (openSectionModule M (D none)) := by
change Subsingleton Γ(M,D none)
exact FiniteSupport.sections_subsingleton_of_stalks
((SheafOfModules.toSheaf X.ringCatSheaf).obj M) _ (fun x hx => hM x hx)
letI : Subsingleton (openSectionModule N (D none)) := by
change Subsingleton Γ(N,D none)
exact FiniteSupport.sections_subsingleton_of_stalks
((SheafOfModules.toSheaf X.ringCatSheaf).obj N) _ (fun x hx => hN x hx)
exact LinearEquiv.ofSubsingleton _ _
| some x =>
let hVu : V x ≤ U x := inf_le_left
let nest (P : X.Modules) : (P.restrict (U x).ι).restrict (X.homOfLE hVu) ≅ P.restrict (V x).ι :=
((Scheme.Modules.restrictFunctorComp (X.homOfLE hVu) (U x).ι).app P).symm ≪≫
(Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι hVu)).app P
exact openSectionIso M N (V x) ((nest M).symm ≪≫
(Scheme.Modules.restrictFunctor (X.homOfLE hVu)).mapIso (e x) ≪≫ nest N)
exact (disjointSupportCoverEquiv M D hcover (hz M hM)).trans
((LinearEquiv.piCongrRight eD).trans (disjointSupportCoverEquiv N D hcover (hz N hN)).symm)
end
end PiExponentSeshadri.Geometry
end OAI
|