| import OAI.Algebra.AuslanderReiten.Complete.CompleteTailMap | |
| import OAI.Algebra.AuslanderReiten.Stable.StableExtNaturality | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit | |
| open CompleteComparison | |
| universe u | |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] | |
| variable {P Q : ChainComplex (ModuleCat.{u} R) ℤ} | |
| variable (hP : TotallyAcyclicComplex P) (hQ : TotallyAcyclicComplex Q) | |
| variable (f : P ⟶ Q) (N : ModuleCat.{u} R) | |
| theorem positiveExtStableEquiv_precomp (n : ℕ) | |
| (x : Ext (ModuleCat.of R (Cokernel Q 0)) N (n+1)) : | |
| positiveExtStableEquiv (k := k) hP N n | |
| ((Ext.mk₀ (ModuleCat.ofHom (cokernelMap f 0))).comp x (zero_add _)) = | |
| StableHom.precomp (cokernelMap f ((n+1 : ℕ) : ℤ)) | |
| (positiveExtStableEquiv (k := k) hQ N n x) := by | |
| obtain ⟨g, hg, rfl⟩ := (completeExtResolution hQ).extMk_surjective x (n+2) rfl | |
| rw [ProjectiveResolution.mk₀_comp_extMk g (n+2) rfl hg (completeTailHom f hP hQ), | |
| positiveExtStableEquiv_extMk, positiveExtStableEquiv_extMk] | |
| change StableHom.classOf R _ _ _ = StableHom.precomp _ (StableHom.classOf R _ _ _) | |
| rw [StableHom.precomp_classOf] | |
| apply congrArg (StableHom.classOf R _ _) | |
| apply LinearMap.ext | |
| intro z | |
| obtain ⟨z, rfl⟩ := Submodule.mkQ_surjective | |
| (LinearMap.range (P.d (((n+1 : ℕ) : ℤ)+1) ((n+1 : ℕ) : ℤ)).hom) z | |
| rfl | |
| end ArExplicit | |
| end | |
| end OAI | |