| import OAI.NumberTheory.PiExponent.Approximation.SectionPullback |
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| namespace OAI |
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| namespace PiExponentSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace |
| open PiExponentSeshadri.Frames |
| variable {X Y : Scheme.{0}} |
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| lemma endValue_conjugation (b : O Y ≅ O Y) (a : O Y ⟶ O Y) : |
| endValue (b.hom ≫ a ≫ b.inv) = endValue a := by |
| rw [endValue_comp, endValue_comp] |
| have hb : endValue b.hom * endValue b.inv = 1 := by |
| rw [← endValue_comp, b.hom_inv_id, endValue_id] |
| calc |
| endValue b.hom * (endValue a * endValue b.inv) = |
| endValue a * (endValue b.hom * endValue b.inv) := by ring |
| _ = endValue a := by rw [hb, mul_one] |
|
|
| lemma exists_frame_natIso (F G : X.Modules ⥤ Y.Modules) (v : F ≅ G) |
| (uF : F.obj (O X) ≅ O Y) (uG : G.obj (O X) ≅ O Y) |
| {M : X.Modules} (e : G.obj M ≅ O Y) (s : O X ⟶ M) : |
| ∃ eF : F.obj M ≅ O Y, |
| coefficient eF (uF.inv ≫ F.map s) = coefficient e (uG.inv ≫ G.map s) := by |
| let b : O Y ≅ O Y := uF.symm ≪≫ v.app (O X) ≪≫ uG |
| refine ⟨v.app M ≪≫ e ≪≫ b.symm, ?_⟩ |
| have h : (uF.inv ≫ F.map s) ≫ (v.app M).hom = |
| b.hom ≫ (uG.inv ≫ G.map s) := by |
| simp only [b, Iso.trans_hom, Iso.symm_hom, Category.assoc] |
| simp only [Iso.hom_inv_id_assoc] |
| exact congrArg (fun q => uF.inv ≫ q) (v.hom.naturality s) |
| change endValue (((uF.inv ≫ F.map s) ≫ (v.app M).hom) ≫ e.hom ≫ b.inv) = _ |
| rw [h] |
| change endValue (b.hom ≫ (uG.inv ≫ G.map s ≫ e.hom) ≫ b.inv) = _ |
| exact endValue_conjugation b _ |
|
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| def pullbackRestrictNatIso (f : Y ⟶ X) (U : X.Opens) : |
| Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι ≅ |
| Scheme.Modules.restrictFunctor U.ι ⋙ Scheme.Modules.pullback (f ∣_ U) := |
| Functor.isoWhiskerLeft (Scheme.Modules.pullback f) |
| (Scheme.Modules.restrictFunctorIsoPullback (f ⁻¹ᵁ U).ι) ≪≫ |
| Scheme.Modules.pullbackComp (f ⁻¹ᵁ U).ι f ≪≫ |
| Scheme.Modules.pullbackCongr (morphismRestrict_ι f U).symm ≪≫ |
| (Scheme.Modules.pullbackComp (f ∣_ U) U.ι).symm ≪≫ |
| Functor.isoWhiskerRight (Scheme.Modules.restrictFunctorIsoPullback U.ι).symm |
| (Scheme.Modules.pullback (f ∣_ U)) |
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| theorem exists_restricted_pullback_frame (f : Y ⟶ X) (U : X.Opens) |
| {M : X.Modules} (e : M.restrict U.ι ≅ O U.toScheme) (s : O X ⟶ M) : |
| ∃ eF : ((Scheme.Modules.pullback f).obj M).restrict (f ⁻¹ᵁ U).ι ≅ |
| O (f ⁻¹ᵁ U).toScheme, |
| coefficient eF (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) = |
| (f ∣_ U).appTop (coefficient e (restrictSection U.ι s)) := by |
| let F : X.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules := |
| Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι |
| let G : X.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules := |
| Scheme.Modules.restrictFunctor U.ι ⋙ Scheme.Modules.pullback (f ∣_ U) |
| let uF : F.obj (O X) ≅ O (f ⁻¹ᵁ U).toScheme := |
| (Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι).mapIso (pullbackUnitIso f) ≪≫ |
| Scheme.Modules.restrictUnitIso (f ⁻¹ᵁ U).ι |
| let uG : G.obj (O X) ≅ O (f ⁻¹ᵁ U).toScheme := |
| (Scheme.Modules.pullback (f ∣_ U)).mapIso (Scheme.Modules.restrictUnitIso U.ι) ≪≫ |
| pullbackUnitIso (f ∣_ U) |
| obtain ⟨eF,he⟩ := exists_frame_natIso F G (pullbackRestrictNatIso f U) uF uG |
| (pullbackFrame (f ∣_ U) e) s |
| refine ⟨eF, ?_⟩ |
| have he' : coefficient eF (restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s)) = |
| coefficient (pullbackFrame (f ∣_ U) e) |
| (pullbackSection (f ∣_ U) (restrictSection U.ι s)) := by |
| let P : X.Modules ⥤ Y.Modules := Scheme.Modules.pullback f |
| let R : Y.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules := |
| Scheme.Modules.restrictFunctor (f ⁻¹ᵁ U).ι |
| let S : X.Modules ⥤ U.toScheme.Modules := Scheme.Modules.restrictFunctor U.ι |
| let Q : U.toScheme.Modules ⥤ (f ⁻¹ᵁ U).toScheme.Modules := |
| Scheme.Modules.pullback (f ∣_ U) |
| let p : P.obj (O X) ≅ O Y := pullbackUnitIso f |
| let r : R.obj (O Y) ≅ O (f ⁻¹ᵁ U).toScheme := |
| Scheme.Modules.restrictUnitIso (f ⁻¹ᵁ U).ι |
| let a : S.obj (O X) ≅ O U.toScheme := Scheme.Modules.restrictUnitIso U.ι |
| let q : Q.obj (O U.toScheme) ≅ O (f ⁻¹ᵁ U).toScheme := pullbackUnitIso (f ∣_ U) |
| have hl : restrictSection (f ⁻¹ᵁ U).ι (pullbackSection f s) = uF.inv ≫ F.map s := by |
| simp only [restrictSection, pullbackSection] |
| change r.inv ≫ R.map (p.inv ≫ P.map s) = (r.inv ≫ R.map p.inv) ≫ R.map (P.map s) |
| exact (congrArg (fun k => r.inv ≫ k) (R.map_comp p.inv (P.map s))).trans |
| (Category.assoc r.inv (R.map p.inv) (R.map (P.map s))).symm |
| have hr : pullbackSection (f ∣_ U) (restrictSection U.ι s) = uG.inv ≫ G.map s := by |
| change q.inv ≫ Q.map (a.inv ≫ S.map s) = (q.inv ≫ Q.map a.inv) ≫ Q.map (S.map s) |
| exact (congrArg (fun k => q.inv ≫ k) (Q.map_comp a.inv (S.map s))).trans |
| (Category.assoc q.inv (Q.map a.inv) (Q.map (S.map s))).symm |
| exact (congrArg (coefficient eF) hl).trans |
| (he.trans (congrArg (coefficient (pullbackFrame (f ∣_ U) e)) hr).symm) |
| rw [he', coefficient_pullback] |
|
|
| end |
| end PiExponentSeshadri.Geometry |
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| end OAI |
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