def
tensor_product.direct_sum
(R : Type u_1)
[comm_ring R]
(ι₁ : Type u_2)
(ι₂ : Type u_3)
[decidable_eq ι₁]
[decidable_eq ι₂]
(M₁ : ι₁ → Type u_4)
(M₂ : ι₂ → Type u_5)
[Π (i₁ : ι₁), add_comm_group (M₁ i₁)]
[Π (i₂ : ι₂), add_comm_group (M₂ i₂)]
[Π (i₁ : ι₁), module R (M₁ i₁)]
[Π (i₂ : ι₂), module R (M₂ i₂)] :
tensor_product R (direct_sum ι₁ (λ (i₁ : ι₁), M₁ i₁)) (direct_sum ι₂ (λ (i₂ : ι₂), M₂ i₂)) ≃ₗ[R] direct_sum (ι₁ × ι₂) (λ (i : ι₁ × ι₂), tensor_product R (M₁ i.fst) (M₂ i.snd))
The linear equivalence (⨁ i₁, M₁ i₁) ⊗ (⨁ i₂, M₂ i₂) ≃ (⨁ i₁, ⨁ i₂, M₁ i₁ ⊗ M₂ i₂), i.e.
"tensor product distributes over direct sum".
Equations
- tensor_product.direct_sum R ι₁ ι₂ M₁ M₂ = linear_equiv.of_linear (tensor_product.lift (direct_sum.to_module R ι₁ (direct_sum ι₂ (λ (i₂ : ι₂), M₂ i₂) →ₗ[R] direct_sum (ι₁ × ι₂) (λ (i : ι₁ × ι₂), tensor_product R (M₁ i.fst) (M₂ i.snd))) (λ (i₁ : ι₁), (direct_sum.to_module R ι₂ (M₁ i₁ →ₗ[R] direct_sum (ι₁ × ι₂) (λ (i : ι₁ × ι₂), tensor_product R (M₁ i.fst) (M₂ i.snd))) (λ (i₂ : ι₂), (tensor_product.curry (direct_sum.lof R (ι₁ × ι₂) (λ (i : ι₁ × ι₂), tensor_product R (M₁ i.fst) (M₂ i.snd)) (i₁, i₂))).flip)).flip))) (direct_sum.to_module R (ι₁ × ι₂) (tensor_product R (direct_sum ι₁ (λ (i₁ : ι₁), M₁ i₁)) (direct_sum ι₂ (λ (i₂ : ι₂), M₂ i₂))) (λ (i : ι₁ × ι₂), tensor_product.map (direct_sum.lof R ι₁ M₁ i.fst) (direct_sum.lof R ι₂ M₂ i.snd))) _ _
@[simp]
theorem
tensor_product.direct_sum_lof_tmul_lof
(R : Type u_1)
[comm_ring R]
(ι₁ : Type u_2)
(ι₂ : Type u_3)
[decidable_eq ι₁]
[decidable_eq ι₂]
(M₁ : ι₁ → Type u_4)
(M₂ : ι₂ → Type u_5)
[Π (i₁ : ι₁), add_comm_group (M₁ i₁)]
[Π (i₂ : ι₂), add_comm_group (M₂ i₂)]
[Π (i₁ : ι₁), module R (M₁ i₁)]
[Π (i₂ : ι₂), module R (M₂ i₂)]
(i₁ : ι₁)
(m₁ : M₁ i₁)
(i₂ : ι₂)
(m₂ : M₂ i₂) :
⇑(tensor_product.direct_sum R ι₁ ι₂ M₁ M₂) (⇑(direct_sum.lof R ι₁ M₁ i₁) m₁ ⊗ₜ[R] ⇑(direct_sum.lof R ι₂ M₂ i₂) m₂) = ⇑(direct_sum.lof R (ι₁ × ι₂) (λ (i : ι₁ × ι₂), tensor_product R (M₁ i.fst) (M₂ i.snd)) (i₁, i₂)) (m₁ ⊗ₜ[R] m₂)