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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 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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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