openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /FrameSubopensCoherence.lean
| import OAI.NumberTheory.PiExponent.Approximation.FrameSubopens | |
| namespace OAI | |
| namespace PiExponentSeshadri.Frames | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite | |
| variable {X : Scheme} {M : X.Modules} {U V W : X.Opens} | |
| lemma restrictOpenFrame_app (h : V ≤ U) (e : M.restrict U.ι ≅ O U.toScheme) | |
| (A : V.toScheme.Opens) (x : Γ(M.restrict V.ι,A)) : | |
| (restrictOpenFrame h e).hom.app A x = | |
| X.presheaf.map (homOfLE (show V.ι ''ᵁ A ≤ U.ι ''ᵁ (X.homOfLE h ''ᵁ A) from by | |
| simp [← Scheme.Hom.comp_image])).op | |
| (e.hom.app (X.homOfLE h ''ᵁ A) | |
| (M.presheaf.map (homOfLE | |
| (show U.ι ''ᵁ (X.homOfLE h ''ᵁ A) ≤ V.ι ''ᵁ A from by | |
| simp [← Scheme.Hom.comp_image])).op x)) := by | |
| simp only [restrictOpenFrame, Iso.trans_hom, Iso.symm_hom, | |
| Scheme.Modules.Hom.comp_app, Iso.app_inv, Iso.app_hom, | |
| Scheme.Modules.restrictFunctorCongr_inv_app_app, | |
| Scheme.Modules.restrictFunctorComp_hom_app_app] | |
| change ((X.homOfLE h).appIso A).hom (e.hom.app (X.homOfLE h ''ᵁ A) | |
| ((M.presheaf.map _ ≫ M.presheaf.map _) x)) = _ | |
| simp only [Scheme.Hom.appIso_hom', Scheme.homOfLE_appLE] | |
| erw [← Functor.map_comp] | |
| rfl | |
| lemma restrictOpenFrame_trans (h : W ≤ V) (k : V ≤ U) | |
| (e : M.restrict U.ι ≅ O U.toScheme) : | |
| restrictOpenFrame h (restrictOpenFrame k e) = restrictOpenFrame (h.trans k) e := by | |
| apply Iso.ext | |
| ext A x | |
| have hh := restrictOpenFrame_app h (restrictOpenFrame k e) A x | |
| let y := M.presheaf.map (homOfLE | |
| (show V.ι ''ᵁ (X.homOfLE h ''ᵁ A) ≤ W.ι ''ᵁ A from by | |
| simp [← Scheme.Hom.comp_image])).op x | |
| have hk := restrictOpenFrame_app k e (X.homOfLE h ''ᵁ A) y | |
| have hhk := congrArg (X.presheaf.map (homOfLE | |
| (show W.ι ''ᵁ A ≤ V.ι ''ᵁ (X.homOfLE h ''ᵁ A) from by | |
| simp [← Scheme.Hom.comp_image])).op) hk | |
| have hc := restrictOpenFrame_app (h.trans k) e A x | |
| refine hh.trans (hhk.trans (Eq.trans ?_ hc.symm)) | |
| change (X.presheaf.map _ ≫ X.presheaf.map _) | |
| (e.hom.app (X.homOfLE k ''ᵁ (X.homOfLE h ''ᵁ A)) | |
| ((M.presheaf.map _ ≫ M.presheaf.map _) x)) = | |
| X.presheaf.map _ (e.hom.app (X.homOfLE (h.trans k) ''ᵁ A) | |
| (M.presheaf.map _ x)) | |
| erw [← Functor.map_comp, ← Functor.map_comp] | |
| let B := X.homOfLE k ''ᵁ (X.homOfLE h ''ᵁ A) | |
| let C := X.homOfLE (h.trans k) ''ᵁ A | |
| have hA : C ≤ B := by | |
| simp only [B, C, ← Scheme.Hom.comp_image, Scheme.homOfLE_homOfLE] | |
| exact le_rfl | |
| have hB : U.ι ''ᵁ B ≤ W.ι ''ᵁ A := by | |
| simp [B, ← Scheme.Hom.comp_image] | |
| have hC : W.ι ''ᵁ A ≤ U.ι ''ᵁ C := by | |
| simp [C, ← Scheme.Hom.comp_image] | |
| have hn := CategoryTheory.congr_fun | |
| (e.hom.mapPresheaf.naturality (homOfLE hA).op) | |
| (M.presheaf.map (homOfLE hB).op x) | |
| change e.hom.app C ((M.restrict U.ι).presheaf.map (homOfLE hA).op | |
| (M.presheaf.map (homOfLE hB).op x)) = | |
| U.toScheme.presheaf.map (homOfLE hA).op | |
| (e.hom.app B (M.presheaf.map (homOfLE hB).op x)) at hn | |
| refine Eq.trans ?_ (Eq.trans (congrArg (X.presheaf.map (homOfLE hC).op) hn).symm ?_) | |
| · change X.presheaf.map _ (e.hom.app B (M.presheaf.map _ x)) = | |
| (X.presheaf.map _ ≫ X.presheaf.map _) (e.hom.app B (M.presheaf.map _ x)) | |
| erw [← Functor.map_comp] | |
| rfl | |
| · apply congrArg (X.presheaf.map (homOfLE hC).op) | |
| apply congrArg (e.hom.app C) | |
| change (M.presheaf.map _ ≫ M.presheaf.map _) x = M.presheaf.map _ x | |
| erw [← Functor.map_comp] | |
| rfl | |
| end | |
| end PiExponentSeshadri.Frames | |
| end OAI | |