openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /CompositePullbackFrame.lean
| import OAI.NumberTheory.PiExponent.Ampleness.ClosedAmpleRestriction | |
| import OAI.NumberTheory.PiExponent.Polynomials.PullbackFrameCoefficient | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponent.CompositePullbackFrame | |
| open AlgebraicGeometry CategoryTheory | |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames | |
| variable {X Y Z : Scheme.{0}} | |
| private theorem mapIso_trans_inv_comp {C D : Type*} [Category C] [Category D] | |
| (F : C ⥤ D) {A B T : C} {U : D} (e : A ≅ B) (u : F.obj B ≅ U) | |
| (s : A ⟶ T) : | |
| (F.mapIso e ≪≫ u).inv ≫ F.map s = u.inv ≫ F.map (e.inv ≫ s) := by | |
| simp only [Iso.trans_inv, Functor.mapIso_inv, Functor.map_comp, Category.assoc] | |
| private theorem map_comp_iso_inv {C D : Type*} [Category C] [Category D] | |
| (F : C ⥤ D) {A B T : C} {U : D} (u : U ⟶ F.obj A) | |
| (s : A ⟶ B) (e : B ≅ T) : | |
| (u ≫ F.map (s ≫ e.hom)) ≫ (F.mapIso e).inv = u ≫ F.map s := by | |
| simp only [Functor.mapIso_inv, Category.assoc, ← Functor.map_comp, | |
| Iso.hom_inv_id, Category.comp_id] | |
| theorem exists_frame (f : X ⟶ Y) (φ : Z ⟶ X) [IsOpenImmersion φ] | |
| (V : Y.Opens) (g : Z ⟶ V.toScheme) (h : φ ≫ f = g ≫ V.ι) | |
| {M : Y.Modules} (e : M.restrict V.ι ≅ O V.toScheme) : | |
| ∃ E : ((Scheme.Modules.pullback f).obj M).restrict φ ≅ O Z, | |
| ∀ s : O Y ⟶ M, | |
| coefficient E (restrictSection φ (pullbackSection f s)) = | |
| g.appTop (coefficient e (restrictSection V.ι s)) := by | |
| let F := Scheme.Modules.pullback f ⋙ Scheme.Modules.restrictFunctor φ | |
| let G := Scheme.Modules.restrictFunctor V.ι ⋙ Scheme.Modules.pullback g | |
| let v : F ≅ G := | |
| Functor.isoWhiskerLeft (Scheme.Modules.pullback f) | |
| (Scheme.Modules.restrictFunctorIsoPullback φ) ≪≫ | |
| Scheme.Modules.pullbackComp φ f ≪≫ | |
| Scheme.Modules.pullbackCongr h ≪≫ | |
| (Scheme.Modules.pullbackComp g V.ι).symm ≪≫ | |
| Functor.isoWhiskerRight (Scheme.Modules.restrictFunctorIsoPullback V.ι).symm | |
| (Scheme.Modules.pullback g) | |
| let uF : F.obj (O Y) ≅ O Z := | |
| (Scheme.Modules.restrictFunctor φ).mapIso (pullbackUnitIso f) ≪≫ | |
| Scheme.Modules.restrictUnitIso φ | |
| let uG : G.obj (O Y) ≅ O Z := | |
| (Scheme.Modules.pullback g).mapIso (Scheme.Modules.restrictUnitIso V.ι) ≪≫ | |
| pullbackUnitIso g | |
| obtain ⟨E, hE⟩ := PullbackFrameCoefficient.exists_frame_natIso_all F G v uF uG | |
| (pullbackFrame g e) | |
| refine ⟨E, fun s => ?_⟩ | |
| have hs : coefficient E (restrictSection φ (pullbackSection f s)) = | |
| coefficient (pullbackFrame g e) (pullbackSection g (restrictSection V.ι s)) := by | |
| simp only [restrictSection, pullbackSection] | |
| exact (congrArg (coefficient E) | |
| (mapIso_trans_inv_comp (Scheme.Modules.restrictFunctor φ) | |
| (pullbackUnitIso f) (Scheme.Modules.restrictUnitIso φ) | |
| ((Scheme.Modules.pullback f).map s)).symm).trans | |
| ((hE s).trans (congrArg (coefficient (pullbackFrame g e)) | |
| (mapIso_trans_inv_comp (Scheme.Modules.pullback g) | |
| (Scheme.Modules.restrictUnitIso V.ι) (pullbackUnitIso g) | |
| ((Scheme.Modules.restrictFunctor V.ι).map s)))) | |
| rw [hs, coefficient_pullback] | |
| theorem exists_power_frame (L : LineBundle Y) (f : X ⟶ Y) | |
| (φ : Z ⟶ X) [IsOpenImmersion φ] (V : Y.Opens) | |
| (g : Z ⟶ V.toScheme) (h : φ ≫ f = g ≫ V.ι) (n : ℕ) | |
| (e : (modulePow Y L.sheaf n).restrict V.ι ≅ O V.toScheme) : | |
| ∃ E : (modulePow X (L.pullback f).sheaf n).restrict φ ≅ O Z, | |
| ∀ s : O Y ⟶ modulePow Y L.sheaf n, | |
| coefficient E (restrictSection φ (pullbackPowerSection L f n s)) = | |
| g.appTop (coefficient e (restrictSection V.ι s)) := by | |
| obtain ⟨E, hE⟩ := exists_frame f φ V g h e | |
| refine ⟨(Scheme.Modules.restrictFunctor φ).mapIso | |
| (PiExponentSeshadri.PullbackTensor.powIso f L n).symm ≪≫ E, fun s => ?_⟩ | |
| erw [coefficient_transport] | |
| have hs : restrictSection φ (pullbackPowerSection L f n s) ≫ | |
| ((Scheme.Modules.restrictFunctor φ).mapIso (PiExponentSeshadri.PullbackTensor.powIso f L n).symm).hom = | |
| restrictSection φ (pullbackSection f s) := by | |
| simp only [restrictSection, pullbackPowerSection, Functor.mapIso_hom, Iso.symm_hom] | |
| exact map_comp_iso_inv (Scheme.Modules.restrictFunctor φ) | |
| (Scheme.Modules.restrictUnitIso φ).inv (pullbackSection f s) | |
| (PiExponentSeshadri.PullbackTensor.powIso f L n) | |
| exact (congrArg (coefficient E) hs).trans (hE s) | |
| end PiExponent.CompositePullbackFrame | |
| end | |
| end OAI | |