File size: 1,740 Bytes
ae00908 | 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 43 44 45 46 47 | import OAI.Algebra.Finitistic.Tensor.BalancedTensor
namespace OAI
noncomputable section
namespace LittleFinitistic.BalancedTensor
universe u v w w' w'' z
variable {k : Type u} [Field k] {A : Type v} [Ring A] [Algebra k A]
variable {M : Type w} [AddCommGroup M] [Module k M] [Module Aᵐᵒᵖ M]
variable {M' : Type w'} [AddCommGroup M'] [Module k M'] [Module Aᵐᵒᵖ M']
variable {M'' : Type w''} [AddCommGroup M''] [Module k M''] [Module Aᵐᵒᵖ M'']
variable (N : Type z) [AddCommGroup N] [Module k N] [Module A N]
variable [IsScalarTower k Aᵐᵒᵖ M] [IsScalarTower k Aᵐᵒᵖ M']
[IsScalarTower k Aᵐᵒᵖ M'']
/-- Tensor an actual right-module homomorphism with the identity. -/
def mapLeft (f : M →ₗ[Aᵐᵒᵖ] M') : Tensor k A M N →ₗ[k] Tensor k A M' N :=
lift ((mk k A M' N).compl₁₂ (f.restrictScalars k) LinearMap.id) (by
intro a m n
change mk k A M' N (f (MulOpposite.op a • m)) n = mk k A M' N (f m) (a • n)
rw [map_smul]
exact balance a (f m) n)
@[simp] theorem mapLeft_mk (f : M →ₗ[Aᵐᵒᵖ] M') (m : M) (n : N) :
mapLeft (k := k) N f (mk k A M N m n) = mk k A M' N (f m) n := rfl
@[simp] theorem mapLeft_id : mapLeft (k := k) N (LinearMap.id : M →ₗ[Aᵐᵒᵖ] M) = LinearMap.id := by
apply ext
intro m n
rfl
@[simp] theorem mapLeft_comp (f : M →ₗ[Aᵐᵒᵖ] M') (g : M' →ₗ[Aᵐᵒᵖ] M'') :
mapLeft (k := k) N (g.comp f) = (mapLeft (k := k) N g).comp (mapLeft (k := k) N f) := by
apply ext
intro m n
rfl
@[simp] theorem mapLeft_add (f g : M →ₗ[Aᵐᵒᵖ] M') :
mapLeft (k := k) N (f+g) = mapLeft (k := k) N f + mapLeft (k := k) N g := by
apply ext
intro m n
simp
end LittleFinitistic.BalancedTensor
end
end OAI
|