| import OAI.Algebra.AuslanderReiten.HomComplex.FullHomShift | |
| import OAI.Algebra.AuslanderReiten.Cones.CokernelShift | |
| import OAI.Algebra.AuslanderReiten.Complete.CompleteShiftDual | |
| import OAI.Algebra.AuslanderReiten.Stable.StableHomEquiv | |
| namespace OAI | |
| open CategoryTheory | |
| open CochainComplex CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.CompleteComparison | |
| universe u | |
| variable {k R : Type u} [CommRing k] [Ring R] [Algebra k R] | |
| noncomputable def fullHomStableZeroLinearEquiv | |
| (K L : CochainComplex (ModuleCat.{u} R) ℤ) | |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| (hL : ExactProjectiveComplex (chainOfCochain L)) : | |
| CohomologyClass K L 0 ≃ₗ[k] | |
| StableHom R (Cokernel (chainOfCochain K) 0) (Cokernel (chainOfCochain L) 0) := | |
| (fullHomStableLinearEquiv K L 0 hK (hL.cochainShift 0)).trans | |
| (StableHom.targetEquiv (k := k) (shiftedCokernelEquiv L 0 0 rfl)) | |
| noncomputable def fullHomSourceStableLinearEquiv | |
| (K L : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) | |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| (hL : ExactProjectiveComplex (chainOfCochain L)) : | |
| CohomologyClass K L n ≃ₗ[k] | |
| StableHom R (Cokernel (chainOfCochain K) n) (Cokernel (chainOfCochain L) 0) := | |
| (leftShiftCohomologyEquiv k K L n (-n) 0 (by omega)).trans | |
| ((fullHomStableZeroLinearEquiv (K⟦-n⟧) L (hK.cochainShift (-n)) hL).trans | |
| (StableHom.sourceEquiv (k := k) (shiftedCokernelEquiv K (-n) n rfl).symm)) | |
| end ArExplicit.CompleteComparison | |
| end OAI | |