| import OAI.Algebra.AuslanderReiten.Stable.StableExtEquiv | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| namespace ArExplicit.CompleteComparison | |
| universe u | |
| variable {R : Type u} [Ring R] | |
| variable {P Q : ChainComplex (ModuleCat.{u} R) ℤ} (f : P ⟶ Q) | |
| def nonnegativeTailMap : nonnegativeTail P ⟶ nonnegativeTail Q where | |
| f n := f.f (n : ℤ) | |
| comm' i j h := by | |
| change j+1=i at h | |
| subst i | |
| rw [nonnegativeTail_d, nonnegativeTail_d] | |
| exact f.comm _ _ | |
| theorem nonnegativeTailMap_augmentation : | |
| (nonnegativeTailMap f).f 0 ≫ (nonnegativeTailAugmentation Q).f 0 = | |
| (nonnegativeTailAugmentation P).f 0 ≫ ModuleCat.ofHom (cokernelMap f 0) := by | |
| rw [nonnegativeTailAugmentation_f_zero, nonnegativeTailAugmentation_f_zero] | |
| rfl | |
| def completeTailHom (hP : TotallyAcyclicComplex P) (hQ : TotallyAcyclicComplex Q) : | |
| ProjectiveResolution.Hom (completeExtResolution hP) (completeExtResolution hQ) | |
| (ModuleCat.ofHom (cokernelMap f 0)) where | |
| hom := nonnegativeTailMap f | |
| hom_f_zero_comp_π_f_zero := by | |
| erw [ChainComplex.single₀_map_f_zero] | |
| exact nonnegativeTailMap_augmentation f | |
| end ArExplicit.CompleteComparison | |
| end | |
| end OAI | |