| import OAI.Algebra.AuslanderReiten.Ext.ExtIsoCongr | |
| import Mathlib.Algebra.Homology.DerivedCategory.Ext.Map | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits | |
| namespace ArExplicit | |
| universe u v w | |
| variable {k : Type u} [CommRing k] | |
| variable {C : Type v} [Category.{u} C] [Abelian C] [HasExt.{u} C] [Linear k C] | |
| variable {D : Type w} [Category.{u} D] [Abelian D] [HasExt.{u} D] [Linear k D] | |
| variable {X Y Z : C} | |
| theorem selfExtIso_trans (e : X ≅ Y) (f : Y ≅ Z) (n : ℕ) (x : Ext X X n) : | |
| selfExtIso (k := k) f n (selfExtIso (k := k) e n x) = | |
| selfExtIso (k := k) (e ≪≫ f) n x := by | |
| simp [selfExtIso_apply] | |
| theorem selfExtIso_symm_apply (e : X ≅ Y) (n : ℕ) (x : Ext Y Y n) : | |
| (selfExtIso (k := k) e n).symm x = selfExtIso (k := k) e.symm n x := by | |
| apply (selfExtIso (k := k) e n).injective | |
| simp [selfExtIso_apply] | |
| theorem selfExtIso_eqToIso (h : X = Y) (n : ℕ) (x : Ext X X n) : | |
| selfExtIso (k := k) (eqToIso h) n x = h ▸ x := by | |
| subst h | |
| simp | |
| variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F] [PreservesFiniteColimits F] | |
| theorem selfExtIso_mapExactFunctor (e : X ≅ Y) (n : ℕ) (x : Ext X X n) : | |
| (selfExtIso (k := k) e n x).mapExactFunctor F = | |
| selfExtIso (k := k) (F.mapIso e) n (x.mapExactFunctor F) := by | |
| simp only [selfExtIso_apply, Ext.mapExactFunctor_comp, Ext.mapExactFunctor_mk₀, | |
| Functor.mapIso_hom, Functor.mapIso_inv] | |
| theorem selfExtIso_conjugated_functor | |
| (G : C ⥤ C) [G.Additive] [PreservesFiniteLimits G] [PreservesFiniteColimits G] | |
| (e : Y ≅ X) (E : Y ≅ G.obj Y) (n : ℕ) (x : Ext X X n) : | |
| selfExtIso (k := k) e n | |
| (selfExtIso (k := k) E.symm n | |
| ((selfExtIso (k := k) e.symm n x).mapExactFunctor G)) = | |
| selfExtIso (k := k) (e.symm ≪≫ E ≪≫ G.mapIso e).symm n | |
| (x.mapExactFunctor G) := by | |
| rw [selfExtIso_mapExactFunctor, selfExtIso_trans, selfExtIso_trans] | |
| have heq : G.mapIso e.symm ≪≫ E.symm ≪≫ e = | |
| (e.symm ≪≫ E ≪≫ G.mapIso e).symm := by | |
| apply Iso.ext | |
| simp | |
| rw [heq] | |
| end ArExplicit | |
| end | |
| end OAI | |