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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 | import OAI.NumberTheory.PiExponent.Polynomials.FrameCoefficients
namespace OAI
namespace PiExponentSeshadri.Frames
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
variable {X : Scheme} {M : X.Modules}
def restrictOpenFrame {U V : X.Opens} (h : V ≤ U) (e : M.restrict U.ι ≅ O U.toScheme) :
M.restrict V.ι ≅ O V.toScheme :=
((Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M).symm ≪≫
(Scheme.Modules.restrictFunctorComp (X.homOfLE h) U.ι).app M ≪≫
restrictFrame (X.homOfLE h) e
lemma restrictOpenFrame_coefficient_global {U V : X.Opens} (h : V ≤ U)
(e : M.restrict U.ι ≅ O U.toScheme) (s : O X ⟶ M) :
coefficient (restrictOpenFrame h e) (restrictSection V.ι s) =
(X.homOfLE h).appTop (coefficient e (restrictSection U.ι s)) := by
unfold restrictOpenFrame
rw [coefficient_transport]
rw [← restrictSection_congr (X.homOfLE_ι h) s, Category.assoc]
have he : (Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).hom.app M ≫
((Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M).symm.hom = 𝟙 _ :=
Iso.hom_inv_id ((Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M)
rw [he, Category.comp_id]
exact coefficient_restrict_comp (X.homOfLE h) U.ι e s
lemma frameChange_restrictOpenFrame {U V W : X.Opens} (hWU : W ≤ U) (hWV : W ≤ V)
(e : M.restrict U.ι ≅ O U.toScheme) (f : M.restrict V.ι ≅ O V.toScheme)
(s : O X ⟶ M) (hs : coefficient e (restrictSection U.ι s) = 1) :
(frameChange (restrictOpenFrame hWU e) (restrictOpenFrame hWV f) : Γ(W.toScheme,⊤)) =
(X.homOfLE hWV).appTop (coefficient f (restrictSection V.ι s)) := by
have h := coefficient_change (restrictOpenFrame hWU e) (restrictOpenFrame hWV f)
(restrictSection W.ι s)
rw [restrictOpenFrame_coefficient_global, restrictOpenFrame_coefficient_global,
hs, map_one, mul_one] at h
exact h.symm
end
end PiExponentSeshadri.Frames
end OAI
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