| import OAI.Algebra.AuslanderReiten.Complete.CompleteStableComparison |
| import Mathlib.Algebra.Homology.Embedding.ExtendHomotopy |
| import Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology |
|
|
| namespace OAI |
|
|
| open CategoryTheory |
|
|
| namespace ArExplicit.CompleteComparison |
|
|
| universe u |
|
|
| variable {R : Type u} [Ring R] |
|
|
| def negIndexEmbedding : (ComplexShape.up ℤ).Embedding (ComplexShape.down ℤ) := |
| ComplexShape.Embedding.mk' _ _ (fun i : ℤ => -i) neg_injective (by |
| intro i j |
| change i + 1 = j ↔ -j + 1 = -i |
| omega) |
| |
| instance : negIndexEmbedding.IsRelIff := by |
| dsimp [negIndexEmbedding] |
| infer_instance |
| |
| noncomputable abbrev chainOfCochain (K : CochainComplex (ModuleCat.{u} R) ℤ) := |
| K.extend negIndexEmbedding |
| |
| private instance negIndexFunctor_additive : |
| (negIndexEmbedding.extendHomotopyFunctor (ModuleCat.{u} R)).Additive := by |
| have := Functor.additive_of_iso |
| (negIndexEmbedding.extendHomotopyFunctorFactors (ModuleCat.{u} R)).symm |
| exact Functor.additive_of_full_essSurj_comp |
| (HomotopyCategory.quotient (ModuleCat.{u} R) (ComplexShape.up ℤ)) _ |
| |
| noncomputable def fullHomStableEquiv |
| (K L : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) |
| (hL : ExactProjectiveComplex (chainOfCochain (L⟦n⟧))) : |
| CochainComplex.HomComplex.CohomologyClass K L n ≃+ |
| StableHom R (Cokernel (chainOfCochain K) 0) (Cokernel (chainOfCochain (L⟦n⟧)) 0) := |
| CochainComplex.HomComplex.CohomologyClass.homAddEquiv.trans |
| ((AddEquiv.ofBijective |
| (Functor.mapAddHom (negIndexEmbedding.extendHomotopyFunctor (ModuleCat.{u} R))) |
| ⟨Functor.map_injective _, Functor.map_surjective _⟩).trans |
| (homotopyCokernelEquiv hK hL)) |
| |
| theorem fullHomStableEquiv_eq_zero_iff |
| (K L : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) |
| (hK : TotallyAcyclicComplex (chainOfCochain K)) |
| (hL : ExactProjectiveComplex (chainOfCochain (L⟦n⟧))) |
| (x : CochainComplex.HomComplex.CohomologyClass K L n) : |
| fullHomStableEquiv K L n hK hL x = 0 ↔ x = 0 := |
| map_eq_zero_iff (fullHomStableEquiv K L n hK hL) (fullHomStableEquiv K L n hK hL).injective |
| |
| end ArExplicit.CompleteComparison |
| |
| end OAI |
| |
