| import OAI.NumberTheory.PiExponent.Geometry.ProjectiveFrameNaturality |
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| namespace OAI |
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| namespace PiExponentSeshadri.ProjectiveChartSections |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite |
| open PiExponentSeshadri.Geometry ModuleFlasque RestrictionCohomology Frames |
| variable {X : Scheme} (U : X.Opens) (M : X.Modules) |
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| lemma framedCoefficientsEquiv_smul |
| (e : M.restrict U.ι ≅ structureSheaf U.toScheme) |
| (a : Γ(X,U)) (s : Γ(M,U)) : |
| framedCoefficientsEquiv U M e (a • s) = |
| a * framedCoefficientsEquiv U M e s := by |
| have hs : (restrictionSectionsIso U M).inv (a • s) = |
| U.topIso.inv a • (restrictionSectionsIso U M).inv s := by |
| change M.presheaf.map (eqToHom U.ι_image_top).op (a • s) = |
| (U.ι.appIso ⊤).inv (U.topIso.inv a) • |
| M.presheaf.map (eqToHom U.ι_image_top).op s |
| rw [M.map_smul, U.ι_appIso] |
| rfl |
| change U.topIso.hom (e.hom.app ⊤ ((restrictionSectionsIso U M).inv (a • s))) = |
| a * U.topIso.hom (e.hom.app ⊤ ((restrictionSectionsIso U M).inv s)) |
| rw [hs, e.hom.app_smul] |
| change U.topIso.hom (U.topIso.inv a * _) = _ |
| erw [map_mul] |
| erw [Iso.inv_hom_id_apply] |
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| lemma framedCoefficientsEquiv_change |
| (e f : M.restrict U.ι ≅ structureSheaf U.toScheme) (s : Γ(M,U)) : |
| framedCoefficientsEquiv U M f s = |
| U.topIso.hom (frameChange e f : Γ(U.toScheme,⊤)) * |
| framedCoefficientsEquiv U M e s := by |
| let t := (restrictionSectionsIso U M).inv s |
| have he : e.inv.app ⊤ (e.hom.app ⊤ t) = t := by |
| exact congrArg (fun q => q.app ⊤ t) e.hom_inv_id |
| have h := end_apply (e.inv ≫ f.hom) ⊤ (e.hom.app ⊤ t) |
| change f.hom.app ⊤ (e.inv.app ⊤ (e.hom.app ⊤ t)) = |
| (show Γ(U.toScheme,⊤) from e.hom.app ⊤ t) * |
| (frameChange e f : Γ(U.toScheme,⊤)) at h |
| rw [he, mul_comm] at h |
| change U.topIso.hom (f.hom.app ⊤ t) = |
| U.topIso.hom (frameChange e f : Γ(U.toScheme,⊤)) * U.topIso.hom (e.hom.app ⊤ t) |
| rw [h, map_mul] |
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| end |
| end PiExponentSeshadri.ProjectiveChartSections |
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| end OAI |
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