openAI-math-dataset / lean /OAI /Algebra /AuslanderReiten /Tensor /EnvelopingCoefficientComplex.lean
| import OAI.Algebra.AuslanderReiten.Tensor.EnvelopingEvaluationCoefficient | |
| import OAI.Algebra.AuslanderReiten.Ext.ExternalEvaluationTerms | |
| namespace OAI | |
| noncomputable section | |
| namespace ArExplicit.EnvelopingBimodule | |
| open CategoryTheory | |
| open scoped TensorProduct ModuleCat.Algebra | |
| universe u | |
| variable (k R : Type u) [Field k] [Ring R] [Algebra k R] | |
| variable {X Y : ModuleCat.{u} R} | |
| variable [Module k X] [IsScalarTower k R X] [Module k Y] [IsScalarTower k R Y] | |
| variable (e : X ≅ Y) (P : ChainComplex (ModuleCat.{u} (R ⊗[k] Rᵐᵒᵖ)) ℕ) | |
| def coefficientComplexIso : | |
| ((evaluationFunctor k R X).mapHomologicalComplex (ComplexShape.down ℕ)).obj P ≅ | |
| ((evaluationFunctor k R Y).mapHomologicalComplex (ComplexShape.down ℕ)).obj P := | |
| (NatIso.mapHomologicalComplex (evaluationCoefficientIso k R e) (ComplexShape.down ℕ)).app P | |
| theorem coefficientComplexIso_comparison (m n : ℕ) (f : P.X m ⟶ P.X n) : | |
| (evaluationFunctor k R X).map f ≫ (coefficientComplexIso k R e P).hom.f n = | |
| (coefficientComplexIso k R e P).hom.f m ≫ (evaluationFunctor k R Y).map f := | |
| (evaluationCoefficientIso k R e).hom.naturality f | |
| end ArExplicit.EnvelopingBimodule | |
| end | |
| end OAI | |