| import OAI.Algebra.AuslanderReiten.Ext.ExtIsoFunctor | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| namespace ArExplicit | |
| 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 : C ⥤ C) [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F] | |
| variable {X Y : C} (e : X ≅ Y) (E : Y ≅ F.obj Y) | |
| theorem exactFunctor_scalar_transport (n : ℕ) (c : k) | |
| (h : ∀ z : Ext Y Y n, selfExtIso (k := k) E.symm n (z.mapExactFunctor F) = c • z) | |
| (x : Ext X X n) : | |
| selfExtIso (k := k) (e ≪≫ E ≪≫ F.mapIso e.symm).symm n (x.mapExactFunctor F) = c • x := by | |
| have ht := selfExtIso_conjugated_functor (k := k) F e.symm E n x | |
| simp only [Iso.symm_symm_eq] at ht | |
| rw [← ht, h, map_smul] | |
| congr 1 | |
| simp [selfExtIso_apply] | |
| end ArExplicit | |
| end | |
| end OAI | |