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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | 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
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