| import OAI.Algebra.AuslanderReiten.Ext.ExactFunctorExtNaturality | |
| import OAI.Algebra.AuslanderReiten.Ext.ExtIsoFunctor | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| namespace ArExplicit.ResolutionProducts | |
| universe u v | |
| variable {k : Type u} [CommRing k] | |
| variable {C : Type v} [Category.{u} C] [Abelian C] [HasExt.{u} C] [Linear k C] | |
| variable (F G : C ⥤ C) [F.Additive] [G.Additive] | |
| [PreservesFiniteLimits F] [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] | |
| [PreservesFiniteLimits G] [PreservesFiniteColimits G] [G.PreservesProjectiveObjects] | |
| variable (η : F ≅ G) {X : C} (P : ProjectiveResolution X) | |
| include P in | |
| theorem mapExactFunctor_natIso (n : ℕ) (x : Ext X X n) : | |
| selfExtIso (k := k) (η.app X) n (x.mapExactFunctor F) = x.mapExactFunctor G := by | |
| rw [selfExtIso_apply, Ext.comp_assoc_of_third_deg_zero] | |
| simp only [Iso.app_hom, Iso.app_inv] | |
| rw [mapExactFunctor_natural F G η.hom P] | |
| simp | |
| include P in | |
| theorem exactFunctor_action_natIso (E : X ≅ G.obj X) (n : ℕ) (x : Ext X X n) : | |
| selfExtIso (k := k) (E ≪≫ (η.app X).symm).symm n (x.mapExactFunctor F) = | |
| selfExtIso (k := k) E.symm n (x.mapExactFunctor G) := by | |
| rw [← mapExactFunctor_natIso (k := k) F G η P, selfExtIso_trans] | |
| rfl | |
| end ArExplicit.ResolutionProducts | |
| end | |
| end OAI | |