| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.ChartModuleMap |
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| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.Geometry.BaseSections |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite |
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| variable {K : Type} [CommRing K] {X : Scheme.{0}} |
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| lemma congr_apply (k : K →+* Γ(X,⊤)) (M : X.Modules) {U V : X.Opens} |
| (h : U = V) (m : Sections k M U) : |
| congr k M h m = M.presheaf.map (eqToHom h.symm).op m := by |
| subst V |
| change m = M.presheaf.map (𝟙 (op U)) m |
| exact congrArg (fun f : M.presheaf.obj (op U) ⟶ M.presheaf.obj (op U) => f m) |
| (M.presheaf.map_id (op U)).symm |
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| def nestedRestriction (M : X.Modules) {U W : X.Opens} (h : W ≤ U) : |
| (M.restrict U.ι).restrict (X.homOfLE h) ≅ M.restrict W.ι := |
| ((Scheme.Modules.restrictFunctorComp (X.homOfLE h) U.ι).app M).symm ≪≫ |
| (Scheme.Modules.restrictFunctorCongr (X.homOfLE_ι h)).app M |
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| lemma nestedBase (k : K →+* Γ(X,⊤)) {U W : X.Opens} (h : W ≤ U) : |
| (X.homOfLE h).appTop.hom.comp (U.ι.appTop.hom.comp k) = W.ι.appTop.hom.comp k := by |
| rw [← RingHom.comp_assoc, ← CommRingCat.hom_comp, ← Scheme.Hom.comp_appTop, |
| Scheme.homOfLE_ι] |
|
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| lemma chartTop_nested (k : K →+* Γ(X,⊤)) (M : X.Modules) |
| {U W : X.Opens} (h : W ≤ U) |
| (m : Sections (U.ι.appTop.hom.comp k) (M.restrict U.ι) ⊤) : |
| chartTop k M W (chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k) |
| (X.homOfLE h) (nestedBase k h) (M.restrict U.ι) (nestedRestriction M h) m) = |
| res k M h (chartTop k M U m) := by |
| simp only [chartTop, LinearEquiv.trans_apply, restrictEquiv, congr_apply, |
| chartMap, chartModuleMap, LinearMap.coe_comp, |
| chartModuleEquiv, moduleIsoSections, openImmersionTop, restrictedOpenSections, |
| nestedRestriction, Iso.trans_hom, Iso.symm_hom, Iso.app_hom] |
| change (M.presheaf.map _ ≫ M.presheaf.map _ ≫ M.presheaf.map _ ≫ |
| M.presheaf.map _) m = (M.presheaf.map _ ≫ M.presheaf.map _) m |
| simp only [← M.presheaf.map_comp] |
| apply congrArg (fun morphism : |
| M.presheaf.obj (op (U.ι ''ᵁ (⊤ : U.toScheme.Opens))) ⟶ M.presheaf.obj (op W) => |
| morphism m) |
| exact (M.presheaf.map_comp _ _).symm.trans |
| ((congrArg M.presheaf.map (Subsingleton.elim _ _)).trans |
| (M.presheaf.map_comp _ _)) |
| lemma chartMap_range_coherent (k : K →+* Γ(X,⊤)) (M : X.Modules) |
| {U W : X.Opens} (h : W ≤ U) : |
| (chartMap (U.ι.appTop.hom.comp k) (W.ι.appTop.hom.comp k) (X.homOfLE h) |
| (nestedBase k h) (M.restrict U.ι) (nestedRestriction M h)).range.map |
| (chartTop k M W).toLinearMap = (res k M h).range := by |
| ext x |
| constructor |
| · rintro ⟨y,⟨z,rfl⟩,rfl⟩ |
| exact ⟨chartTop k M U z, (chartTop_nested k M h z).symm⟩ |
| · rintro ⟨z,rfl⟩ |
| refine ⟨_, ⟨(chartTop k M U).symm z,rfl⟩, ?_⟩ |
| exact (chartTop_nested k M h _).trans |
| (congrArg (res k M h) ((chartTop k M U).apply_symm_apply z)) |
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|
| end |
| end MaximalSeshadri.Geometry.BaseSections |
| end |
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| end |
| end |
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| end OAI |
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