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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 39 40 | 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
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