openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Triangular /ConvertedTensorCriterion.lean
| 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 | |