| import OAI.Algebra.AuslanderReiten.Cones.ConeCriterion |
| import OAI.Algebra.AuslanderReiten.Triangular.TriangularCochain |
| import OAI.Algebra.AuslanderReiten.Triangular.ConvertedStructure |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit |
|
|
| open CategoryTheory CompleteComparison TriangularInduction |
| open CochainComplex CochainComplex.HomComplex |
|
|
| universe u |
|
|
| variable {k : Type u} [Field k] (q : k) (B : Type u) |
| variable [AddCommGroup B] [Module k B] [Module (A k q) B] [Module (A k q)ᵐᵒᵖ B] |
| variable [SMulCommClass k (A k q) B] [SMulCommClass (A k q) (A k q)ᵐᵒᵖ B] |
| variable [IsScalarTower k (A k q) B] [IsScalarTower k (A k q)ᵐᵒᵖ B] |
| variable [SMulCommClass k (A k q)ᵐᵒᵖ B] |
| variable [Module.Finite k B] |
| variable [Module.Finite (A k q) B] [Module.Projective (A k q) B] |
| variable [Module.Finite (A k q)ᵐᵒᵖ B] [Module.Projective (A k q)ᵐᵒᵖ B] |
|
|
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
|
|
| variable (P : CochainComplex (ModuleCat.{u} (A k q)) ℤ) |
| variable (φ : jCochain (k := k) (F := B) P ⟶ eCochain (k := k) (F := B) P) |
|
|
| theorem convertedCone_mainConclusions (hq : q ≠ 0) (hq1 : 1 + q ≠ 0) |
| (hP : TotallyAcyclicCochain P) |
| (hinj : ∀ n : ℕ, 0 < n → Function.Injective (coneComparison φ (n : ℤ))) |
| (hsurj : ∀ n : ℕ, Function.Surjective (coneComparison φ (n : ℤ))) |
| (hker : ∃ w : CohomologyClass (eCochain (k := k) (F := B) P) |
| (eCochain (k := k) (F := B) P) 0 × |
| CohomologyClass (jCochain (k := k) (F := B) P) |
| (jCochain (k := k) (F := B) P) 0, |
| w ≠ 0 ∧ coneComparison φ 0 w = 0) : |
| MainConclusions k (ConvertedAlgebra k q B) (ConeModule φ) := by |
| have hJ := jCochain_totallyAcyclic (k := k) (F := B) P hP |
| have hE := eCochain_totallyAcyclic (k := k) (F := B) P hP |
| obtain ⟨hnp, hgp, hself, hreg⟩ := coneModule_properties (k := k) φ hJ hE |
| (eToJCochain_eq_zero k (A k q) B P P) hinj hsurj hker |
| have hC := (mappingCone_totallyAcyclic φ hJ hE).to_chain |
| let := hC.finite 0 |
| let : Module.Finite k ((chainOfCochain (mappingCone φ)).X 0) := |
| Module.Finite.trans (ConvertedAlgebra k q B) _ |
| exact ⟨inferInstance, inferInstance, hnp, hgp, hself, hreg, |
| ⟨ConvertedAlgebra.semisimpleQuotientEquiv q B⟩, |
| ConvertedAlgebra.jacobson_fourth_ne_bot q B hq hq1⟩ |
|
|
| end ArExplicit |
|
|
| end |
|
|
| end OAI |
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|
