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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 | 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
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