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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | 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
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