File size: 2,052 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 | import OAI.NumberTheory.PiExponent.Geometry.ProjectiveFrameNaturality
namespace OAI
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)
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]
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]
end
end PiExponentSeshadri.ProjectiveChartSections
end OAI
|