| import OAI.Algebra.AuslanderReiten.HomComplex.PositiveFullHomProducts | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Abelian | |
| open CochainComplex CochainComplex.HomComplex | |
| namespace ArExplicit.CompleteComparison | |
| universe u | |
| variable {k R : Type u} [Field k] [Ring R] [Algebra k R] [CharP k 2] | |
| variable [HasDerivedCategory (ModuleCat.{u} R)] | |
| variable (K : CochainComplex (ModuleCat.{u} R) ℤ) | |
| variable (hK : TotallyAcyclicComplex (chainOfCochain K)) | |
| theorem fullPositiveExtEquiv_augmented (n : ℕ) (x : Cocycle K K ((n+1 : ℕ) : ℤ)) : | |
| fullPositiveExtEquiv (k := k) K K hK hK.toExactProjectiveComplex n (CohomologyClass.mk x) = | |
| (completeExtResolution hK).extMk | |
| (ResolutionProducts.natComparisonAugmentation (completeExtResolution hK) (n+1) | |
| (cochainTailMap K K (n+1) x.val)) (n+2) rfl | |
| (ResolutionProducts.natComparison_augmented_cycle (completeExtResolution hK) (n+1) | |
| (cochainTailMap K K (n+1) x.val) (cochainTailMap_comm (k := k) K K (n+1) x)) := by | |
| rw [fullPositiveExtEquiv_mk] | |
| exact ResolutionProducts.natComparison_endExt_eq (completeExtResolution hK) (n+1) | |
| (cochainTailMap K K (n+1) x.val) (cochainTailMap_comm (k := k) K K (n+1) x) | |
| end ArExplicit.CompleteComparison | |
| end | |
| end OAI | |