| import OAI.NumberTheory.PiExponent.Polynomials.FrameCoefficients |
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| namespace OAI |
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| namespace PiExponentSeshadri.Frames |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| variable {X : Scheme} {M : X.Modules} |
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|
| 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 |
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| end OAI |
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