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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 32 33 34 35 36 37 38 | 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
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