| import OAI.Algebra.AuslanderReiten.Tensor.BalancedTensorCategory | |
| import Mathlib.CategoryTheory.Linear.LinearFunctor | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory | |
| open scoped ModuleCat.Algebra | |
| namespace ArExplicit.BalancedTensor | |
| universe u | |
| variable {k R S F : Type u} [Field k] [Ring R] [Ring S] [Algebra k R] [Algebra k S] | |
| variable [AddCommGroup F] [Module k F] [Module Rᵐᵒᵖ F] [Module S F] | |
| [SMulCommClass k S F] [SMulCommClass S Rᵐᵒᵖ F] | |
| [IsScalarTower k Rᵐᵒᵖ F] [IsScalarTower k S F] | |
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower | |
| instance functor_linear : | |
| (functor (k := k) (R := R) (S := S) (F := F)).Linear k where | |
| map_smul := by | |
| intro M N g a | |
| apply ModuleCat.hom_ext | |
| apply leftHom_ext | |
| intro f m | |
| change tmul (k := k) (R := R) f ((algebraMap k R a) • g.hom m) = | |
| (algebraMap k S a) • tmul (k := k) (R := R) f (g.hom m) | |
| rw [IsScalarTower.algebraMap_smul R a (g.hom m), | |
| IsScalarTower.algebraMap_smul S a (tmul (k := k) (R := R) f (g.hom m))] | |
| exact map_smul (mk (k := k) (R := R) f) a (g.hom m) | |
| end ArExplicit.BalancedTensor | |
| end | |
| end OAI | |