File size: 1,481 Bytes
216fb31 | 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 | import OAI.Algebra.Finitistic.Dimension.PDBalancedRight
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
namespace OAI
noncomputable section
namespace LittleFinitistic.PDTensor
open CategoryTheory
universe u
variable (k A : Type u) [Field k] [Ring A] [Algebra k A]
abbrev ScalarModule (M : ModuleCat.{u} A) :=
(ModuleCat.restrictScalars (algebraMap k A)).obj M
instance scalarModuleTower (M : ModuleCat.{u} A) :
IsScalarTower k A (ScalarModule k A M) where
smul_assoc c a m := by
change (c • a) • (m : M) = algebraMap k A c • (a • (m : M))
rw [Algebra.smul_def, mul_smul]
variable (U : Type u) [AddCommGroup U] [Module k U] [Module Aᵐᵒᵖ U]
/-- Tensor with a fixed right module, on actual left modules over A. -/
def tensorFunctor : ModuleCat.{u} A ⥤ ModuleCat.{u} k where
obj M := ModuleCat.of k (BalancedTensor.Tensor k A U (ScalarModule k A M))
map {M N} f := ModuleCat.ofHom
(mapRight (k := k) (M := ScalarModule k A M) (N := ScalarModule k A N) U f.hom)
map_id M := by apply ModuleCat.hom_ext; apply BalancedTensor.ext; intros; rfl
map_comp f g := by apply ModuleCat.hom_ext; apply BalancedTensor.ext; intros; rfl
instance tensorFunctorAdditive : (tensorFunctor k A U).Additive where
map_add := by
intro M N f g
apply ModuleCat.hom_ext
apply BalancedTensor.ext
intro u m
exact map_add (BalancedTensor.mk k A U (ScalarModule k A N) u) (f.hom m) (g.hom m)
end LittleFinitistic.PDTensor
end
end OAI
|