| 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 |
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|
