openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Resolutions /MapResolutionExtConjugate.lean
| import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionExtMk | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| namespace ArExplicit.ResolutionProducts | |
| universe u v | |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C] | |
| variable (F : C ⥤ C) [F.Additive] [PreservesFiniteLimits F] | |
| [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] | |
| variable {X Y : C} (P : ProjectiveResolution X) {n : ℕ} | |
| theorem map_extMk_conjugate (a : X ⟶ F.obj X) (b : F.obj Y ⟶ Y) | |
| (φ : ProjectiveResolution.Hom P (mapResolution F P) a) | |
| (f : P.complex.X n ⟶ Y) (hf : P.complex.d (n+1) n ≫ f = 0) | |
| (g : P.complex.X n ⟶ Y) (hg : P.complex.d (n+1) n ≫ g = 0) | |
| (hgf : g = (φ.hom.f n ≫ F.map f) ≫ b) : | |
| ((Ext.mk₀ a).comp ((P.extMk f (n+1) rfl hf).mapExactFunctor F) (zero_add n)).comp | |
| (Ext.mk₀ b) (add_zero n) = | |
| P.extMk g (n+1) rfl hg := by | |
| subst g | |
| erw [map_extMk, ProjectiveResolution.mk₀_comp_extMk _ _ _ _ φ, | |
| ProjectiveResolution.extMk_comp_mk₀] | |
| end ArExplicit.ResolutionProducts | |
| end | |
| end OAI | |