| import OAI.NumberTheory.PiExponent.Approximation.CocycleTensor |
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| namespace OAI |
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| namespace PiExponentSeshadri.LineBundleGluing |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite MonoidalCategory |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.TensorPure |
| variable {X : Scheme} {ι : Type} {U : ι → X.Opens} |
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| lemma tensorHom_restrict_isIso (W : X.Opens) |
| {P P' Q Q' : PresheafOfModules X.ringCatSheaf.obj} (f : P ⟶ P') (g : Q ⟶ Q') |
| [IsIso ((modulePresheafRestrict W.ι).map f)] |
| [IsIso ((modulePresheafRestrict W.ι).map g)] : |
| IsIso ((modulePresheafRestrict W.ι).map |
| (PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf) f g)) := by |
| let R := modulePresheafRestrict W.ι |
| let : MonoidalCategory (PresheafOfModules W.toScheme.ringCatSheaf.obj) := |
| PresheafOfModulesOfCommRing.monoidalCategory (R := W.toScheme.presheaf) |
| have hh : IsIso (R.map (PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf) f g) ≫ |
| (modulePresheafTensorRestrict W P' Q').hom) := by |
| rw [modulePresheafTensorRestrict_natural] |
| change IsIso (_ ≫ (R.map f ⊗ₘ R.map g)) |
| infer_instance |
| let hc : IsIso (C := PresheafOfModules W.toScheme.ringCatSheaf.obj) |
| (modulePresheafTensorRestrict W P' Q').hom := |
| (modulePresheafTensorRestrict W P' Q').isIso_hom |
| exact (@isIso_comp_right_iff (PresheafOfModules W.toScheme.ringCatSheaf.obj) |
| _ _ _ _ _ _ hc).mp hh |
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| def cocycleTensorComparison (c d : Cocycle U) (hcover : ⊤ ≤ ⨆ i, U i) : |
| (PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).obj |
| (PresheafOfModulesOfCommRing.Monoidal.tensorObj (R := X.presheaf) |
| (presheaf c) (presheaf d)) ≅ moduleTensor X (sheaf c) (sheaf d) := by |
| let f := PresheafOfModulesOfCommRing.Monoidal.tensorHom (R := X.presheaf) |
| ((adj X).unit.app (presheaf c)) ((adj X).unit.app (presheaf d)) |
| have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map f) := by |
| apply sheafification_isIso_of_chart f hcover |
| intro i |
| have := cocycleUnit_restrict_isIso c i |
| have := cocycleUnit_restrict_isIso d i |
| exact tensorHom_restrict_isIso (U i) _ _ |
| exact @asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map f) ht |
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| def trivialSheafIso (hcover : ⊤ ≤ ⨆ i, U i) : |
| sheaf (Cocycle.one (U := U)) ≅ structureSheaf X := by |
| have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map |
| (trivialCoefficientMap (U := U))) := |
| sheafification_isIso_of_chart _ hcover trivialCoefficientMap_restrict_isIso |
| exact (@asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map |
| (trivialCoefficientMap (U := U))) ht).symm ≪≫ |
| (asIso (adj X).counit).app (structureSheaf X) |
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| def tensorInverseIso (c : Cocycle U) (hcover : ⊤ ≤ ⨆ i, U i) : |
| moduleTensor X (sheaf c) (sheaf c.inverse) ≅ structureSheaf X := by |
| have ht : IsIso ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map |
| (inverseProductMap c)) := |
| sheafification_isIso_of_chart _ hcover (inverseProductMap_restrict_isIso c) |
| exact (cocycleTensorComparison c c.inverse hcover).symm ≪≫ |
| @asIso _ _ _ _ ((PresheafOfModules.sheafification (𝟙 X.ringCatSheaf.obj)).map |
| (inverseProductMap c)) ht ≪≫ |
| trivialSheafIso hcover |
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| end |
| end PiExponentSeshadri.LineBundleGluing |
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| end OAI |
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