openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /CompositePullbackCoefficient.lean
| import OAI.NumberTheory.PiExponent.Polynomials.PullbackFrameCoefficient | |
| namespace OAI | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory | |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames | |
| namespace PiExponent.CompositePullbackCoefficient | |
| theorem exists_frame {X Y Z : Scheme.{0}} (f : X ⟶ Y) (g : Y ⟶ Z) | |
| {M : Z.Modules} (e : (Scheme.Modules.pullback (f ≫ g)).obj M ≅ O X) : | |
| ∃ e' : (Scheme.Modules.pullback f).obj ((Scheme.Modules.pullback g).obj M) ≅ O X, | |
| ∀ s : O Z ⟶ M, | |
| coefficient e' (pullbackSection f (pullbackSection g s)) = | |
| coefficient e (pullbackSection (f ≫ g) s) := by | |
| let F := Scheme.Modules.pullback g ⋙ Scheme.Modules.pullback f | |
| let G := Scheme.Modules.pullback (f ≫ g) | |
| let uF : F.obj (O Z) ≅ O X := | |
| (Scheme.Modules.pullback f).mapIso (pullbackUnitIso g) ≪≫ pullbackUnitIso f | |
| obtain ⟨e', he'⟩ := PullbackFrameCoefficient.exists_frame_natIso_all | |
| F G (Scheme.Modules.pullbackComp f g) uF (pullbackUnitIso (f ≫ g)) e | |
| refine ⟨e', fun s => ?_⟩ | |
| simpa only [F, G, uF, pullbackSection, Iso.trans_inv, Functor.mapIso_inv, | |
| Functor.comp_map, Functor.map_comp, Category.assoc] using he' s | |
| end PiExponent.CompositePullbackCoefficient | |
| end | |
| end OAI | |