File size: 9,365 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 | import OAI.NumberTheory.PiExponent.Approximation.ModuleLinePowerLaws
import OAI.NumberTheory.PiExponent.Geometry.LineBundleGluingTransitions
namespace OAI
noncomputable section
namespace PiExponentSeshadri.LineBundleGluing
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite MonoidalCategory
open PiExponentSeshadri.Geometry PiExponentSeshadri.TensorPure
variable {X : Scheme} {ι : Type} {U : ι → X.Opens}
def Cocycle.one : Cocycle U where
transition _ _ _ _ _ := 1
restriction _ _ _ _ _ := by simp
identity _ _ _ := rfl
cocycle _ _ _ _ _ _ _ := one_mul _
def Cocycle.inverse (c : Cocycle U) : Cocycle U where
transition i j W hi hj := (c.transition i j W hi hj)⁻¹
restriction := by
intro i j V W h hi hj
have h' : restrictUnit h (c.transition i j _ hi hj) =
c.transition i j _ (h.trans hi) (h.trans hj) := Units.ext (c.restriction i j h hi hj)
exact congrArg Units.val ((map_inv (restrictUnit h) _).symm.trans (congrArg Inv.inv h'))
identity i W hi := by rw [c.identity, inv_one]
cocycle i j k W hi hj hk := by rw [← mul_inv, c.cocycle]
def inverseProduct (c : Cocycle U) (V : X.Opens)
(s : sections c V) (t : sections c.inverse V) : sections (Cocycle.one (U := U)) V :=
⟨fun p => s.val p * t.val p, by
constructor
· intro i W W' hW hi h
change X.presheaf.map (homOfLE h).op (_ * _) = _
rw [map_mul, s.property.1 i W W' hW hi h, t.property.1 i W W' hW hi h]
· intro i j W hW hi hj
change s.val ⟨i,W,hW,hi⟩ * t.val ⟨i,W,hW,hi⟩ = 1 * _
rw [s.property.2 i j W hW hi hj, t.property.2 i j W hW hi hj, one_mul]
change (_ * _) * ((↑((c.transition i j W hi hj)⁻¹ : Γ(X,W)ˣ) : Γ(X,W)) * _) = _
rw [mul_mul_mul_comm, Units.mul_inv, one_mul]⟩
def inverseProductMap (c : Cocycle U) :
PresheafOfModulesOfCommRing.Monoidal.tensorObj (R := X.presheaf)
(presheaf c) (presheaf c.inverse) ⟶ presheaf (Cocycle.one (U := U)) where
app V := ModuleCat.MonoidalCategory.tensorLift (inverseProduct c V.unop)
(by intros; apply Subtype.ext; funext p; exact add_mul _ _ _)
(by intros; apply Subtype.ext; funext p; exact mul_assoc _ _ _)
(by intros; apply Subtype.ext; funext p; exact mul_add _ _ _)
(by intros; apply Subtype.ext; funext p; exact mul_left_comm _ _ _)
naturality {V W} f := by
apply ModuleCat.MonoidalCategory.tensor_ext
intro s t
rfl
def trivialCoefficientMap : (structureSheaf X).val ⟶ presheaf (Cocycle.one (U := U)) where
app V := ModuleCat.ofHom (R := Γ(X, V.unop))
(X := ModuleCat.of Γ(X, V.unop) Γ(X, V.unop))
(Y := ModuleCat.of Γ(X, V.unop) (sections (Cocycle.one (U := U)) V.unop)) {
toFun a := ⟨fun p => X.presheaf.map (homOfLE p.2.property.1).op a, by
constructor
· intro i W W' hW hi h
change (X.presheaf.map _ ≫ X.presheaf.map _) a = _
rw [← Functor.map_comp]
rfl
· intros; exact (one_mul _).symm⟩
map_add' a b := by apply Subtype.ext; funext p; exact map_add _ _ _
map_smul' a b := by apply Subtype.ext; funext p; exact map_mul _ _ _ }
naturality {V W} f := by
apply ModuleCat.hom_ext
apply LinearMap.ext
intro a
apply Subtype.ext
funext p
change (X.presheaf.map _ ≫ X.presheaf.map _) a = _
rw [← Functor.map_comp]
rfl
lemma cocycleUnit_restrict_isIso (c : Cocycle U) (i : ι) :
IsIso ((modulePresheafRestrict (U i).ι).map ((adj X).unit.app (presheaf c))) := by
let R : PresheafOfModules X.ringCatSheaf.obj ⥤
PresheafOfModules (U i).toScheme.ringCatSheaf.obj := modulePresheafRestrict (U i).ι
have hη : IsIso ((adj (U i).toScheme).unit.app (R.obj (presheaf c))) :=
sheafification_unit_isIso_of_iso _ (structureSheaf (U i).toScheme)
(chartPresheafFrame c i)
have hh : IsIso (R.map ((adj X).unit.app (presheaf c)) ≫
((moduleSheafificationRestrict (U i).ι).hom.app (presheaf c)).val) := by
erw [sheafify_restrict_unit]
exact hη
let V : (U i).toScheme.Modules ⥤ PresheafOfModules (U i).toScheme.ringCatSheaf.obj :=
SheafOfModules.forget (U i).toScheme.ringCatSheaf
let hv : IsIso (C := PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
(((moduleSheafificationRestrict (U i).ι).hom.app (presheaf c)).val) :=
(V.mapIso ((moduleSheafificationRestrict (U i).ι).app (presheaf c))).isIso_hom
exact (@isIso_comp_right_iff (PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
_ _ _ _ _ _ hv).mp hh
lemma trivialCoefficientMap_frame (i : ι) :
(modulePresheafRestrict (U i).ι).map (trivialCoefficientMap (U := U)) ≫
(chartPresheafFrame (Cocycle.one (U := U)) i).hom =
(Scheme.Modules.restrictUnitIso (U i).ι).hom.val := by
ext V a
change X.presheaf.map (homOfLE (show (U i).ι ''ᵁ V.unop ≤ (U i).ι ''ᵁ V.unop from le_rfl)).op a = _
simp only [homOfLE_refl, op_id, CategoryTheory.Functor.map_id]
change a = ((U i).ι.appIso V.unop).hom a
simp only [Scheme.Opens.ι_appIso, Iso.refl_hom]
rfl
lemma trivialCoefficientMap_restrict_isIso (i : ι) :
IsIso ((modulePresheafRestrict (U i).ι).map (trivialCoefficientMap (U := U))) := by
let V : (U i).toScheme.Modules ⥤ PresheafOfModules (U i).toScheme.ringCatSheaf.obj :=
SheafOfModules.forget (U i).toScheme.ringCatSheaf
have hv : IsIso (C := PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
((Scheme.Modules.restrictUnitIso (U i).ι).hom.val) :=
(V.mapIso (Scheme.Modules.restrictUnitIso (U i).ι)).isIso_hom
have hh : IsIso ((modulePresheafRestrict (U i).ι).map (trivialCoefficientMap (U := U)) ≫
(chartPresheafFrame (Cocycle.one (U := U)) i).hom) := by
rw [trivialCoefficientMap_frame]
exact hv
let hc : IsIso (C := PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
(chartPresheafFrame (Cocycle.one (U := U)) i).hom :=
(chartPresheafFrame (Cocycle.one (U := U)) i).isIso_hom
exact (@isIso_comp_right_iff (PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
_ _ _ _ _ _ hc).mp hh
lemma inverseProductMap_frame (c : Cocycle U) (i : ι) :
(modulePresheafRestrict (U i).ι).map (inverseProductMap c) ≫
(chartPresheafFrame (Cocycle.one (U := U)) i).hom =
(modulePresheafTensorRestrict (U i) (presheaf c) (presheaf c.inverse)).hom ≫
PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := (U i).toScheme.presheaf)
(chartPresheafFrame c i).hom (chartPresheafFrame c.inverse i).hom ≫
(by
let : MonoidalCategory (PresheafOfModules (U i).toScheme.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := (U i).toScheme.presheaf)
exact (λ_ (structureSheaf (U i).toScheme).val).hom) := by
let : MonoidalCategory (PresheafOfModules (U i).toScheme.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := (U i).toScheme.presheaf)
let l := (modulePresheafRestrict (U i).ι).map (inverseProductMap c) ≫
(chartPresheafFrame (Cocycle.one (U := U)) i).hom
let r := (modulePresheafTensorRestrict (U i) (presheaf c) (presheaf c.inverse)).hom ≫
PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := (U i).toScheme.presheaf)
(chartPresheafFrame c i).hom (chartPresheafFrame c.inverse i).hom ≫
(λ_ (structureSheaf (U i).toScheme).val).hom
change l = r
ext V : 1
apply ModuleCat.hom_ext
apply LinearMap.ext
intro z
induction z using TensorProduct.inductionOn with
| tmul s t => rfl
| add a b ha hb =>
exact ((l.app V).hom.map_add a b).trans
((congrArg₂ (· + ·) ha hb).trans ((r.app V).hom.map_add a b).symm)
lemma inverseProductMap_restrict_isIso (c : Cocycle U) (i : ι) :
IsIso ((modulePresheafRestrict (U i).ι).map (inverseProductMap c)) := by
let : MonoidalCategory (PresheafOfModules (U i).toScheme.ringCatSheaf.obj) :=
PresheafOfModulesOfCommRing.monoidalCategory (R := (U i).toScheme.presheaf)
let e := modulePresheafTensorRestrict (U i) (presheaf c) (presheaf c.inverse) ≪≫
tensorIso (chartPresheafFrame c i) (chartPresheafFrame c.inverse i) ≪≫
λ_ (structureSheaf (U i).toScheme).val
have hh : IsIso ((modulePresheafRestrict (U i).ι).map (inverseProductMap c) ≫
(chartPresheafFrame (Cocycle.one (U := U)) i).hom) := by
rw [inverseProductMap_frame]
exact e.isIso_hom
let hc : IsIso (C := PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
(chartPresheafFrame (Cocycle.one (U := U)) i).hom :=
(chartPresheafFrame (Cocycle.one (U := U)) i).isIso_hom
exact (@isIso_comp_right_iff (PresheafOfModules (U i).toScheme.ringCatSheaf.obj)
_ _ _ _ _ _ hc).mp hh
lemma sheafification_isIso_of_chart {P Q : PresheafOfModules X.ringCatSheaf.obj}
(f : P ⟶ Q) (hcover : ⊤ ≤ ⨆ i, U i)
(hf : ∀ i, IsIso ((modulePresheafRestrict (U i).ι).map f)) :
IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map f) := by
apply PiExponent.NumericalAmpleness.isIso_of_locally_isIso
intro x
obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (hcover (show x ∈ (⊤ : X.Opens) from trivial))
refine ⟨U i, hi, ?_⟩
apply (NatIso.isIso_map_iff (moduleSheafificationRestrict (U i).ι) f).mpr
have := hf i
change IsIso ((PresheafOfModules.sheafification (𝟙 (U i).toScheme.ringCatSheaf.obj)).map
((modulePresheafRestrict (U i).ι).map f))
infer_instance
end PiExponentSeshadri.LineBundleGluing
end
end OAI
|