| import OAI.Algebra.AuslanderReiten.Resolutions.ResolutionExtSquare | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian | |
| namespace ArExplicit | |
| universe u v | |
| variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C] | |
| variable {k : Type v} [CommRing k] [Linear k C] | |
| variable {B S N : C} (P : ProjectiveResolution B) (Q : ProjectiveResolution N) | |
| theorem resolutionExt_cover_conjugate (e : B ≅ S) (j : S ≅ N) | |
| (φ : ProjectiveResolution.Hom P Q (e.hom ≫ j.hom)) (n : ℕ) | |
| (f : P.complex.X n ⟶ S) (g : Q.complex.X n ⟶ N) | |
| (hf : P.complex.d (n+1) n ≫ f = 0) (hg : Q.complex.d (n+1) n ≫ g = 0) | |
| (h : φ.hom.f n ≫ g = f ≫ j.hom) : | |
| selfExtIso (k := k) j n ((Ext.mk₀ e.inv).comp | |
| (P.extMk f (n+1) rfl hf) (zero_add _)) = Q.extMk g (n+1) rfl hg := by | |
| apply selfExtIso_eq_of_square | |
| have he := resolutionExt_square P Q (e.hom ≫ j.hom) j.hom φ n f g hf hg h | |
| have hh := congrArg (fun z => (Ext.mk₀ e.inv).comp z (zero_add n)) he | |
| rw [Ext.mk₀_comp_mk₀_assoc, Iso.inv_hom_id_assoc] at hh | |
| simpa only [Ext.comp_assoc_of_third_deg_zero] using hh | |
| end ArExplicit | |
| end | |
| end OAI | |