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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 | import OAI.Algebra.AuslanderReiten.HomComplex.FiniteFullHomZero
import OAI.Algebra.AuslanderReiten.Stable.TensorStableEnd
import OAI.Algebra.AuslanderReiten.Cones.ConcreteTwoConeEvaluation
namespace OAI
noncomputable section
open CategoryTheory
open CochainComplex CochainComplex.HomComplex
open scoped ModuleCat.Algebra
namespace ArExplicit.A
universe u
variable {k : Type u} [Field k] [CharP k 2] (q : k)
def simpleComplete := (simpleFiniteModule q).completeCochain (AFrobeniusTrace k q)
def fullHomZeroScalarEquiv (hq1 : 1+q ≠ 0) :
CohomologyClass (simpleComplete q) (simpleComplete q) 0 ≃ₗ[k] k :=
(FiniteLeftModule.fullHomZeroEquiv (AFrobeniusTrace k q)
(simpleFiniteModule q) (simpleFiniteModule q)).trans (stableEndEquiv q hq1)
@[simp]
theorem fullHomZeroScalarEquiv_identity (hq1 : 1+q ≠ 0) :
fullHomZeroScalarEquiv q hq1 (CohomologyClass.mk (Cocycle.ofHom (𝟙 (simpleComplete q)))) = 1 := by
change stableEndEquiv q hq1 (FiniteLeftModule.fullHomZeroEquiv (AFrobeniusTrace k q)
(simpleFiniteModule q) (simpleFiniteModule q) _) = 1
erw [FiniteLeftModule.fullHomZeroEquiv_identity, stableEndEquiv_identity]
end ArExplicit.A
end
end OAI
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