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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 | import OAI.Algebra.AuslanderReiten.Triangular.ConvertedCriterion
import OAI.Algebra.AuslanderReiten.Triangular.TriangularCochain
import OAI.Algebra.AuslanderReiten.Triangular.ConvertedStructure
import OAI.Algebra.AuslanderReiten.Triangular.TriangularComparisonTransport
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 (v : P ⟶ tensorCochain (k := k) (F := B) P)
theorem convertedTensorCone_mainConclusions (hq : q ≠ 0) (hq1 : 1 + q ≠ 0)
(hP : TotallyAcyclicCochain P)
(hinj : ∀ n : ℕ, 0 < n → Function.Injective (tensorComparison v (n : ℤ)))
(hsurj : ∀ n : ℕ, Function.Surjective (tensorComparison v (n : ℤ)))
(hker : ∃ w : CohomologyClass P P 0 × CohomologyClass P P 0,
w ≠ 0 ∧ tensorComparison v 0 w = 0) :
MainConclusions k (ConvertedAlgebra k q B) (ConeModule (crossChainMap v)) := by
apply convertedCone_mainConclusions q B P (crossChainMap v) hq hq1 hP
· intro n hn
exact coneComparison_injective_of_tensor v (n : ℤ) (hinj n hn)
· intro n
exact coneComparison_surjective_of_tensor v (n : ℤ) (hsurj n)
· exact coneComparison_nonzero_kernel_of_tensor v 0 hker
end ArExplicit
end
end OAI
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