| import OAI.NumberTheory.PiExponent.Geometry.LineBundleTensor |
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| namespace OAI |
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| universe u |
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| namespace PiExponentSeshadri.Frames |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open scoped AlgebraicGeometry |
| variable {X Y Z : Scheme.{u}} |
| def O (X : Scheme.{u}) : X.Modules := SheafOfModules.unit X.ringCatSheaf |
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| def endValue (f : O X ⟶ O X) : Γ(X, ⊤) := f.app ⊤ (1 : Γ(X, ⊤)) |
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| lemma end_apply (f : O X ⟶ O X) (U : X.Opens) (a : Γ(X, U)) : |
| f.app U a = a * (show Γ(X, U) from f.app U (1 : Γ(X, U))) := by |
| have h := f.app_smul (r := a) (x := (1 : Γ(X, U))) |
| change f.app U (a * 1 : Γ(X, U)) = a * (show Γ(X, U) from f.app U (1 : Γ(X, U))) at h |
| simpa only [mul_one] using h |
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| @[simp] lemma endValue_id : endValue (𝟙 (O X)) = 1 := rfl |
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| lemma endValue_comp (f g : O X ⟶ O X) : |
| endValue (f ≫ g) = endValue f * endValue g := by |
| exact end_apply g ⊤ (endValue f) |
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| def frameChange {M : X.Modules} (e f : M ≅ O X) : Γ(X, ⊤)ˣ where |
| val := endValue (e.inv ≫ f.hom) |
| inv := endValue (f.inv ≫ e.hom) |
| val_inv := by rw [← endValue_comp]; simp |
| inv_val := by rw [← endValue_comp]; simp |
| |
| def coefficient {M : X.Modules} (e : M ≅ O X) (s : O X ⟶ M) : Γ(X, ⊤) := |
| endValue (s ≫ e.hom) |
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| lemma coefficient_change {M : X.Modules} (e f : M ≅ O X) (s : O X ⟶ M) : |
| coefficient f s = (frameChange e f : Γ(X, ⊤)) * coefficient e s := by |
| change endValue (s ≫ f.hom) = endValue (e.inv ≫ f.hom) * endValue (s ≫ e.hom) |
| rw [mul_comm, ← endValue_comp] |
| simp |
|
|
| lemma coefficient_frame {M : X.Modules} (e : M ≅ O X) : coefficient e e.inv = 1 := by |
| simp [coefficient] |
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| def restrictSection (φ : Y ⟶ X) [IsOpenImmersion φ] {M : X.Modules} |
| (s : O X ⟶ M) : O Y ⟶ M.restrict φ := |
| (Scheme.Modules.restrictUnitIso φ).inv ≫ (Scheme.Modules.restrictFunctor φ).map s |
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| def restrictFrame (φ : Y ⟶ X) [IsOpenImmersion φ] {M : X.Modules} |
| (e : M ≅ O X) : M.restrict φ ≅ O Y := |
| (Scheme.Modules.restrictFunctor φ).mapIso e ≪≫ Scheme.Modules.restrictUnitIso φ |
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| lemma end_naturality (f : O X ⟶ O X) (U : X.Opens) : |
| f.app U (1 : Γ(X, U)) = |
| X.presheaf.map (homOfLE (show U ≤ ⊤ from le_top)).op (endValue f) := by |
| have h := CategoryTheory.congr_fun (f.mapPresheaf.naturality |
| (homOfLE (show U ≤ ⊤ from le_top)).op) (1 : Γ(X, ⊤)) |
| change f.app U (X.presheaf.map (homOfLE (show U ≤ ⊤ from le_top)).op (1 : Γ(X, ⊤))) = |
| X.presheaf.map (homOfLE (show U ≤ ⊤ from le_top)).op (endValue f) at h |
| simpa only [map_one] using h |
|
|
| lemma endValue_restrict (φ : Y ⟶ X) [IsOpenImmersion φ] (f : O X ⟶ O X) : |
| endValue (restrictSection φ f ≫ (Scheme.Modules.restrictUnitIso φ).hom) = |
| φ.appTop (endValue f) := by |
| change (φ.appIso ⊤).hom (f.app (φ ''ᵁ ⊤) ((φ.appIso ⊤).inv 1)) = _ |
| rw [map_one, end_naturality] |
| change (X.presheaf.map (homOfLE (show φ ''ᵁ ⊤ ≤ ⊤ from le_top)).op ≫ |
| (φ.appIso ⊤).hom) (endValue f) = _ |
| rw [Scheme.Hom.appIso_hom'] |
| simp only [Scheme.Hom.appTop, Scheme.Hom.appLE, |
| Scheme.Hom.naturality_assoc, ← Functor.map_comp] |
| congr 1 |
| rw [show ((Opens.map φ.base).map (homOfLE (show φ ''ᵁ ⊤ ≤ ⊤ from le_top)).op.unop).op ≫ |
| (homOfLE (show (⊤ : Y.Opens) ≤ φ ⁻¹ᵁ (φ ''ᵁ ⊤) from by intro x _; exact ⟨x, Set.mem_univ _, rfl⟩)).op = 𝟙 _ from Subsingleton.elim _ _] |
| erw [CategoryTheory.Functor.map_id, Category.comp_id] |
| |
| lemma coefficient_restrict (φ : Y ⟶ X) [IsOpenImmersion φ] {M : X.Modules} |
| (e : M ≅ O X) (s : O X ⟶ M) : |
| coefficient (restrictFrame φ e) (restrictSection φ s) = φ.appTop (coefficient e s) := by |
| change endValue ((Scheme.Modules.restrictUnitIso φ).inv ≫ |
| (Scheme.Modules.restrictFunctor φ).map s ≫ |
| (Scheme.Modules.restrictFunctor φ).map e.hom ≫ |
| (Scheme.Modules.restrictUnitIso φ).hom) = _ |
| erw [← Functor.map_comp_assoc] |
| exact endValue_restrict φ (s ≫ e.hom) |
|
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| lemma restrictUnit_comp (φ : Y ⟶ X) (ψ : X ⟶ Z) |
| [IsOpenImmersion φ] [IsOpenImmersion ψ] : |
| (Scheme.Modules.restrictUnitIso (φ ≫ ψ)).inv ≫ |
| (Scheme.Modules.restrictFunctorComp φ ψ).hom.app (O Z) = |
| (Scheme.Modules.restrictUnitIso φ).inv ≫ |
| (Scheme.Modules.restrictFunctor φ).map (Scheme.Modules.restrictUnitIso ψ).inv := by |
| ext U a |
| change Z.presheaf.map (eqToHom (show ψ ''ᵁ (φ ''ᵁ U) = (φ ≫ ψ) ''ᵁ U from by simp)).op |
| ((φ ≫ ψ).appIso U |>.inv |>.hom |>.toFun <| a) = |
| (ψ.appIso (φ ''ᵁ U)).inv ((φ.appIso U).inv a) |
| rw [Scheme.Hom.comp_appIso] |
| simp only [Iso.trans_inv, Functor.mapIso_inv, Iso.op_inv, eqToIso.inv] |
| change ( (φ.appIso U).inv ≫ (ψ.appIso (φ ''ᵁ U)).inv ≫ |
| Z.presheaf.map _ ≫ Z.presheaf.map _) a = _ |
| rw [← CategoryTheory.Functor.map_comp] |
| simp |
| rfl |
|
|
| lemma restrictSection_comp (φ : Y ⟶ X) (ψ : X ⟶ Z) |
| [IsOpenImmersion φ] [IsOpenImmersion ψ] {M : Z.Modules} (s : O Z ⟶ M) : |
| restrictSection (φ ≫ ψ) s ≫ (Scheme.Modules.restrictFunctorComp φ ψ).hom.app M = |
| restrictSection φ (restrictSection ψ s) := by |
| let F := Scheme.Modules.restrictFunctor φ |
| let G := Scheme.Modules.restrictFunctor ψ |
| let H := Scheme.Modules.restrictFunctor (φ ≫ ψ) |
| let e := Scheme.Modules.restrictFunctorComp φ ψ |
| let a : O Y ⟶ H.obj (O Z) := (Scheme.Modules.restrictUnitIso (φ ≫ ψ)).inv |
| let b : O Y ⟶ F.obj (O X) := (Scheme.Modules.restrictUnitIso φ).inv |
| let c : O X ⟶ G.obj (O Z) := (Scheme.Modules.restrictUnitIso ψ).inv |
| change (a ≫ H.map s) ≫ e.hom.app M = b ≫ F.map (c ≫ G.map s) |
| calc |
| _ = a ≫ (H.map s ≫ e.hom.app M) := Category.assoc _ _ _ |
| _ = a ≫ (e.hom.app (O Z) ≫ F.map (G.map s)) := |
| congrArg (fun t => a ≫ t) (e.hom.naturality s) |
| _ = (a ≫ e.hom.app (O Z)) ≫ F.map (G.map s) := (Category.assoc _ _ _).symm |
| _ = (b ≫ F.map c) ≫ F.map (G.map s) := |
| congrArg (fun t => t ≫ F.map (G.map s)) (restrictUnit_comp φ ψ) |
| _ = b ≫ (F.map c ≫ F.map (G.map s)) := Category.assoc _ _ _ |
| _ = _ := congrArg (fun t => b ≫ t) (F.map_comp c (G.map s)).symm |
|
|
| lemma coefficient_transport {M N : X.Modules} (e : M ≅ N) (f : N ≅ O X) |
| (s : O X ⟶ M) : coefficient (e ≪≫ f) s = coefficient f (s ≫ e.hom) := by |
| simp only [coefficient, Iso.trans_hom, Category.assoc] |
|
|
| lemma coefficient_restrict_comp (φ : Y ⟶ X) (ψ : X ⟶ Z) |
| [IsOpenImmersion φ] [IsOpenImmersion ψ] {M : Z.Modules} |
| (e : M.restrict ψ ≅ O X) (s : O Z ⟶ M) : |
| coefficient ((Scheme.Modules.restrictFunctorComp φ ψ).app M ≪≫ restrictFrame φ e) |
| (restrictSection (φ ≫ ψ) s) = φ.appTop (coefficient e (restrictSection ψ s)) := by |
| rw [coefficient_transport] |
| change coefficient (restrictFrame φ e) |
| (restrictSection (φ ≫ ψ) s ≫ (Scheme.Modules.restrictFunctorComp φ ψ).hom.app M) = _ |
| rw [restrictSection_comp, coefficient_restrict] |
|
|
| lemma restrictSection_congr {φ ψ : Y ⟶ X} (h : φ = ψ) |
| [IsOpenImmersion φ] [IsOpenImmersion ψ] {M : X.Modules} (s : O X ⟶ M) : |
| restrictSection φ s ≫ (Scheme.Modules.restrictFunctorCongr h).hom.app M = |
| restrictSection ψ s := by |
| subst ψ |
| have H : (Scheme.Modules.restrictFunctorCongr (show φ = φ from rfl)).hom.app M = 𝟙 _ := by |
| ext U a |
| simp |
| rfl |
| rw [H, Category.comp_id] |
|
|
| lemma overlap_coefficients {W : Scheme.{u}} (φ : Y ⟶ X) (ψ : Z ⟶ X) |
| (a : W ⟶ Y) (b : W ⟶ Z) [IsOpenImmersion φ] [IsOpenImmersion ψ] |
| [IsOpenImmersion a] [IsOpenImmersion b] (h : a ≫ φ = b ≫ ψ) |
| {M : X.Modules} (e : M.restrict φ ≅ O Y) (f : M.restrict ψ ≅ O Z) : |
| ∃ c : Γ(W, ⊤)ˣ, ∀ s : O X ⟶ M, |
| b.appTop (coefficient f (restrictSection ψ s)) = |
| (c : Γ(W, ⊤)) * a.appTop (coefficient e (restrictSection φ s)) := by |
| let E : M.restrict (a ≫ φ) ≅ O W := |
| (Scheme.Modules.restrictFunctorComp a φ).app M ≪≫ restrictFrame a e |
| let F : M.restrict (a ≫ φ) ≅ O W := |
| (Scheme.Modules.restrictFunctorCongr h).app M ≪≫ |
| (Scheme.Modules.restrictFunctorComp b ψ).app M ≪≫ restrictFrame b f |
| refine ⟨frameChange E F, fun s => ?_⟩ |
| have H := coefficient_change E F (restrictSection (a ≫ φ) s) |
| have hE : coefficient E (restrictSection (a ≫ φ) s) = |
| a.appTop (coefficient e (restrictSection φ s)) := coefficient_restrict_comp a φ e s |
| have hF : coefficient F (restrictSection (a ≫ φ) s) = |
| b.appTop (coefficient f (restrictSection ψ s)) := by |
| change coefficient ((Scheme.Modules.restrictFunctorCongr h).app M ≪≫ |
| ((Scheme.Modules.restrictFunctorComp b ψ).app M ≪≫ restrictFrame b f)) _ = _ |
| rw [coefficient_transport] |
| change coefficient _ (restrictSection (a ≫ φ) s ≫ |
| (Scheme.Modules.restrictFunctorCongr h).hom.app M) = _ |
| rw [restrictSection_congr] |
| exact coefficient_restrict_comp b ψ f s |
| rwa [hE, hF] at H |
|
|
| end |
| end PiExponentSeshadri.Frames |
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| end OAI |
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