| 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 | |