| import OAI.Algebra.AuslanderReiten.Stable.SourceStableShift | |
| import OAI.Algebra.AuslanderReiten.Stable.StableExtEquiv | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open CochainComplex CochainComplex.HomComplex | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.CompleteComparison | |
| universe u | |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] | |
| variable (K L : CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| variable (hL : ExactProjectiveComplex (chainOfCochain L)) | |
| def fullPositiveExtEquiv (n : ℕ) : | |
| CohomologyClass K L ((n+1 : ℕ) : ℤ) ≃ₗ[k] | |
| Abelian.Ext (ModuleCat.of R (Cokernel (chainOfCochain K) 0)) | |
| (ModuleCat.of R (Cokernel (chainOfCochain L) 0)) (n+1) := | |
| (fullHomSourceStableLinearEquiv (k := k) K L ((n+1 : ℕ) : ℤ) hK hL).trans | |
| (positiveExtStableEquiv (k := k) hK | |
| (ModuleCat.of R (Cokernel (chainOfCochain L) 0)) n).symm | |
| theorem fullPositiveExtEquiv_stable (n : ℕ) | |
| (x : CohomologyClass K L ((n+1 : ℕ) : ℤ)) : | |
| positiveExtStableEquiv (k := k) hK | |
| (ModuleCat.of R (Cokernel (chainOfCochain L) 0)) n | |
| (fullPositiveExtEquiv (k := k) K L hK hL n x) = | |
| fullHomSourceStableLinearEquiv (k := k) K L ((n+1 : ℕ) : ℤ) hK hL x := | |
| LinearEquiv.apply_symm_apply _ _ | |
| end ArExplicit.CompleteComparison | |
| end | |
| end OAI | |