Google 翻译生成的粗略草稿，质量较低。术语依照术语表，但句子可能有误；之后会由仔细的译文取代。
§ 3。 张量积
1. 张量积两个模
令G 1 , G 2 G_1, G_2 G 1 , G 2 Z \mathbf{Z} Z u ( x 1 , x 2 ) u(x_1, x_2) u ( x 1 , x 2 ) x 1 x_1 x 1 x 2 x_2 x 2 G = G 1 × G 2 G = G_1 \times G_2 G = G 1 × G 2 u u u Z \mathbf{Z} Z Z \mathbf{Z} Z G 1 G_1 G 1 x 1 , y 1 x_1, y_1 x 1 , y 1 G 2 G_2 G 2 x 2 , y 2 x_2, y_2 x 2 , y 2
u ( x 1 + y 1 , x 2 ) = u ( x 1 , x 2 ) + u ( y 1 , x 2 )
u(x_1 + y_1, x_2) = u(x_1, x_2) + u(y_1, x_2)
u ( x 1 + y 1 , x 2 ) = u ( x 1 , x 2 ) + u ( y 1 , x 2 ) u ( x 1 , x 2 + y 2 ) = u ( x 1 , x 2 ) + u ( x 1 , y 2 ) .
u(x_1, x_2 + y_2) = u(x_1, x_2) + u(x_1, y_2).
u ( x 1 , x 2 + y 2 ) = u ( x 1 , x 2 ) + u ( x 1 , y 2 ) . 注这特别意味着 u ( 0 , x 2 ) = u ( x 1 , 0 ) = 0 u(0, x_2) = u(x_1, 0) = 0 u ( 0 , x 2 ) = u ( x 1 , 0 ) = 0 x 1 ∈ G 1 , x 2 ∈ G 2 x_1 \in G_1, x_2 \in G_2 x 1 ∈ G 1 , x 2 ∈ G 2
设 A 为环，E 右A-模，F 左A-模。我们将考虑普遍映射问题（集合理论，IV, § 3, no. 1），其中Σ \Sigma Σ Z \mathbf{Z} Z Z \mathbf{Z} Z α \alpha α E × F E \times F E × F f f f Z \mathbf{Z} Z Z \mathbf{Z} Z x ∈ E , y ∈ F x \in E, y \in F x ∈ E , y ∈ F λ ∈ A \lambda \in A λ ∈ A
f ( x λ , y ) = f ( x , λ y ) .
f(x \lambda, y) = f(x, \lambda y).
f ( x λ , y ) = f ( x , λ y ) . 我们证明这个问题承认解。为此，我们考虑E × F E \times F E × F Z \mathbf{Z} Z C = Z ( E × F ) C = \mathbf{Z}^{(E \times F)} C = Z ( E × F ) Z \mathbf{Z} Z ( x , y ) (x, y) ( x , y ) x ∈ E x \in E x ∈ E y ∈ F y \in F y ∈ F Z \mathbf{Z} Z
{ ( x 1 + x 2 , y ) − ( x 1 , y ) − ( x 2 , y ) ( x , y 1 + y 2 ) − ( x , y 1 ) − ( x , y 2 ) ( x λ , y ) − ( x , λ y )
\begin{cases}
(x_1 + x_2, y) - (x_1, y) - (x_2, y) \\
(x, y_1 + y_2) - (x, y_1) - (x, y_2) \\
(x \lambda, y) - (x, \lambda y)
\end{cases}
⎩ ⎨ ⎧ ( x 1 + x 2 , y ) − ( x 1 , y ) − ( x 2 , y ) ( x , y 1 + y 2 ) − ( x , y 1 ) − ( x , y 2 ) ( x λ , y ) − ( x , λ y ) 其中x , x 1 , x 2 x, x_1, x_2 x , x 1 , x 2 E E E y , y 1 , y 2 y, y_1, y_2 y , y 1 , y 2 F F F λ \lambda λ A A A
右A-模 E 和左A-模 F* 的张量积，用 E ⊗ A F E \otimes_A F E ⊗ A F E ⊗ A F E \otimes_A F E ⊗ A F E ⊗ F E \otimes F E ⊗ F Z \mathbf{Z} Z C / D C/D C / D Z \mathbf{Z} Z E × F E \times F E × F Z \mathbf{Z} Z D D D x ∈ E x \in E x ∈ E y ∈ F y \in F y ∈ F E ⊗ A F E \otimes_A F E ⊗ A F C = Z ( E × F ) C = \mathbf{Z}^{(E \times F)} C = Z ( E × F ) ( x , y ) (x, y) ( x , y ) x ⊗ y x \otimes y x ⊗ y x x x y y y
被以下引用 3 被以下引用 § 3. 。 张量积 , 注 , 命题 1
E × F E \times F E × F E ⊗ A F E \otimes_A F E ⊗ A F ( x , y ) ↦ x ⊗ y (x, y) \mapsto x \otimes y ( x , y ) ↦ x ⊗ y Z \mathbf{Z} Z
我们展示了张量积E ⊗ A F E \otimes_A F E ⊗ A F
(a) *令g g g E ⊗ A F E \otimes_A F E ⊗ A F Z \mathbf{Z} Z G G G Z \mathbf{Z} Z E × F E \times F E × F G G G ( x , y ) ↦ f ( x , y ) = g ( x ⊗ y ) (x, y) \mapsto f(x, y) = g(x \otimes y) ( x , y ) ↦ f ( x , y ) = g ( x ⊗ y ) Z \mathbf{Z} Z
引用了 3 · 被以下引用 8 引用了 II § 1. 。 模 , 命题 17, II § 1 , 定义 1
被以下引用 § 3. 。 张量积 , 推论 , 命题 2 , 命题 3 , 命题 8 , II § 4. 。 关系张量积和同态模之间 , 命题 7, II § 6 , 命题 16, II § 7
(b) 相反，令f f f E × F E \times F E × F Z \mathbf{Z} Z G G G Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F G G G Z \mathbf{Z} Z g g g f ( x , y ) = g ( x ⊗ y ) f(x, y) = g(x \otimes y) f ( x , y ) = g ( x ⊗ y ) x ∈ E , y ∈ F x \in E, y \in F x ∈ E , y ∈ F
如果ϕ \phi ϕ E × F E \times F E × F E ⊗ A F E \otimes_A F E ⊗ A F f = g ∘ ϕ f = g \circ \phi f = g ∘ ϕ f f f C C C Z \mathbf{Z} Z f ˉ \bar{f} f ˉ G G G C C C f ˉ \bar{f} f ˉ D D D C / D = E ⊗ A F C/D = E \otimes_A F C / D = E ⊗ A F G G G Z \mathbf{Z} Z g g g f ˉ = g ∘ ψ \bar{f} = g \circ \psi f ˉ = g ∘ ψ ψ : C → C / D \psi : C \to C/D ψ : C → C / D g g g E ⊗ A F E \otimes_A F E ⊗ A F x ⊗ y x \otimes y x ⊗ y Z \mathbf{Z} Z
将Z \mathbf{Z} Z Z \mathbf{Z} Z 典范同构 将E × F E \times F E × F f f f G G G Z \mathbf{Z} Z H o m Z ( E ⊗ A F , G ) \mathrm{Hom}_{\mathbf{Z}}(E \otimes_A F, G) Hom Z ( E ⊗ A F , G )
当A = Z A = \mathbf{Z} A = Z Z \mathbf{Z} Z f f f C C C D D D
如果现在我们回到一般情况，E ′ E' E ′ F ′ F' F ′ E E E F F F Z \mathbf{Z} Z Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F ( x λ ) ⊗ y − x ⊗ ( λ y ) (x \lambda) \otimes y - x \otimes (\lambda y) ( x λ ) ⊗ y − x ⊗ ( λ y ) Z \mathbf{Z} Z Z \mathbf{Z} Z E ′ ⊗ Z F ′ E' \otimes_{\mathbf{Z}} F' E ′ ⊗ Z F ′ x x x E E E y y y F F F λ \lambda λ A A A
令H H H Z \mathbf{Z} Z h : E × F → H h : E \times F \to H h : E × F → H Z \mathbf{Z} Z H H H h ( E × F ) h(E \times F) h ( E × F ) Z \mathbf{Z} Z G G G E × F E \times F E × F G G G Z \mathbf{Z} Z f f f Z \mathbf{Z} Z g : H → G g : H \to G g : H → G f = g ∘ h f = g \circ h f = g ∘ h ϕ \phi ϕ E × F E \times F E × F E ⊗ A F E \otimes_A F E ⊗ A F E ⊗ A F E \otimes_A F E ⊗ A F θ \theta θ H H H h = θ ∘ ϕ h = \theta \circ \phi h = θ ∘ ϕ
引用了 1 · 被以下引用 1 引用了 E IV § 3. 普遍映射
被以下引用 § 3. 。 张量积
假设 h ( E × F ) h(E \times F) h ( E × F ) H H H g g g
令* E 0 E^0 E 0 F 0 F^0 F 0 E E E F F F A 0 A^0 A 0 则存在一个且仅一个 Z \mathbf{Z} Z σ : E ⊗ A F → F 0 ⊗ A 0 E 0 \sigma : E \otimes_A F \to F^0 \otimes_{A^0} E^0 σ : E ⊗ A F → F 0 ⊗ A 0 E 0 使得 σ ( x ⊗ y ) = y ⊗ x \sigma(x \otimes y) = y \otimes x σ ( x ⊗ y ) = y ⊗ x 对于 x ∈ E x \in E x ∈ E 和 y ∈ F y \in F y ∈ F
被以下引用 3 被以下引用 § 3. 。 张量积 , 推论 , III § 1. 。 代数
由E 0 E^0 E 0 F 0 F^0 F 0 A 0 A^0 A 0 E × F E \times F E × F ( x , y ) ↦ y ⊗ x (x, y) \mapsto y \otimes x ( x , y ) ↦ y ⊗ x F 0 ⊗ A 0 E 0 F^0 \otimes_{A^0} E^0 F 0 ⊗ A 0 E 0 Z \mathbf{Z} Z Z \mathbf{Z} Z σ \sigma σ Z \mathbf{Z} Z τ : F 0 ⊗ A 0 E 0 → E ⊗ A F \tau : F^0 \otimes_{A^0} E^0 \to E \otimes_A F τ : F 0 ⊗ A 0 E 0 → E ⊗ A F τ ( y ⊗ x ) = x ⊗ y \tau(y \otimes x) = x \otimes y τ ( y ⊗ x ) = x ⊗ y σ \sigma σ τ \tau τ
2. 张量积两个线性映射
令A A A E , E ′ E, E' E , E ′ A A A F , F ′ F, F' F , F ′ A A A u : E → E ′ u : E \to E' u : E → E ′ v : F → F ′ v : F \to F' v : F → F ′ A A A
( x , y ) ↦ u ( x ) ⊗ v ( y )
(x, y) \mapsto u(x) \otimes v(y)
( x , y ) ↦ u ( x ) ⊗ v ( y ) E × F E \times F E × F E ′ ⊕ A F ′ E' \oplus_A F' E ′ ⊕ A F ′ Z \mathbf{Z} Z Z \mathbf{Z} Z w : E ⊗ A F → E ′ ⊗ A F ′ w : E \otimes_A F \to E' \otimes_A F' w : E ⊗ A F → E ′ ⊗ A F ′ w ( x ⊗ y ) = u ( x ) ⊗ v ( y )
w(x \otimes y) = u(x) \otimes v(y)
w ( x ⊗ y ) = u ( x ) ⊗ v ( y ) x ∈ E , y ∈ F x \in E, y \in F x ∈ E , y ∈ F u ⊗ v u \otimes v u ⊗ v u u u v v v 从 (3) 可以立即得出，( u , v ) ↦ u ⊗ v (u, v) \mapsto u \otimes v ( u , v ) ↦ u ⊗ v Z -bilinear \mathbf{Z}\text{-bilinear} Z -bilinear
Hom A ( E , E ′ ) × Hom A ( F , F ′ ) → Hom Z ( E ⊗ A F , E ′ ⊗ A F ′ ) .
\operatorname{Hom}_A(E, E') \times \operatorname{Hom}_A(F, F') \to \operatorname{Hom}_{\mathbf{Z}}(E \otimes_A F, E' \otimes_A F').
Hom A ( E , E ′ ) × Hom A ( F , F ′ ) → Hom Z ( E ⊗ A F , E ′ ⊗ A F ′ ) . 通过 no. 1 的命题 1 到它有对应一个 Z \mathbf{Z} Z
(4)
Hom A ( E , E ′ ) ⊗ Z Hom A ( F , F ′ ) → Hom Z ( E ⊗ A F , E ′ ⊗ A F ′ )
\operatorname{Hom}_A(E, E') \otimes_{\mathbf{Z}} \operatorname{Hom}_A(F, F') \to \operatorname{Hom}_{\mathbf{Z}}(E \otimes_A F, E' \otimes_A F')
Hom A ( E , E ′ ) ⊗ Z Hom A ( F , F ′ ) → Hom Z ( E ⊗ A F , E ′ ⊗ A F ′ ) u ⊗ v : E ⊗ A F → E ′ ⊗ A F ′ u \otimes v : E \otimes_A F \to E' \otimes_A F' u ⊗ v : E ⊗ A F → E ′ ⊗ A F ′ u ⊗ v u \otimes v u ⊗ v 不一定是单射或满射 。因此，记号u ⊗ v u \otimes v u ⊗ v 进一步，设E ′ ′ E'' E ′′ A A A F ′ ′ F'' F ′′ A A A u ′ : E ′ → E ′ ′ u' : E' \to E'' u ′ : E ′ → E ′′ v ′ : F ′ → F ′ ′ v' : F' \to F'' v ′ : F ′ → F ′′ A A A
( u ′ ∘ u ) ⊗ ( v ′ ∘ v ) = ( u ′ ⊗ v ′ ) ∘ ( u ⊗ v ) .
(u' \circ u) \otimes (v' \circ v) = (u' \otimes v') \circ (u \otimes v).
( u ′ ∘ u ) ⊗ ( v ′ ∘ v ) = ( u ′ ⊗ v ′ ) ∘ ( u ⊗ v ) . 3. 变换环
令A , B A, B A , B ρ : B → A \rho : B \to A ρ : B → A E E E F F F A A A Z \mathbf{Z} Z
ϕ : ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) → E ⊗ A F
\phi : \rho_*(E) \otimes_B \rho_*(F) \to E \otimes_A F
ϕ : ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) → E ⊗ A F *such that, for all
x ∈ E x \in E x ∈ E and
y ∈ F y \in F y ∈ F , the image under
ϕ \phi ϕ of the element
x ⊗ y x \otimes y x ⊗ y of
ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) \rho_*(E) \otimes_B \rho_*(F) ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) is the element
x ⊗ y x \otimes y x ⊗ y of
E ⊗ A F E \otimes_A F E ⊗ A F ; this
Z \mathbf{Z} Z -linear mapping is surjective.
引用了 1 · 被以下引用 1 引用了 命题 1
被以下引用 推论
我们将ρ ∗ ( E ) × ρ ∗ ( F ) \rho_*(E) \times \rho_*(F) ρ ∗ ( E ) × ρ ∗ ( F ) ( x , y ) ↦ x ⊗ y (x, y) \mapsto x \otimes y ( x , y ) ↦ x ⊗ y E ⊗ A F E \otimes_A F E ⊗ A F Z \mathbf{Z} Z β ∈ B \beta \in B β ∈ B ( x ρ ( β ) ) ⊗ y = x ⊗ ( ρ ( β ) y ) (x \rho(\beta)) \otimes y = x \otimes (\rho(\beta)y) ( x ρ ( β )) ⊗ y = x ⊗ ( ρ ( β ) y ) ϕ \phi ϕ x ⊗ y x \otimes y x ⊗ y Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F
映射 (6) 被称为*规范*。
令J \mathfrak{J} J A A A J \mathfrak{J} J E E E F F F E E E F F F ( A / J ) (A/\mathfrak{J}) ( A / J ) § 1 § 1 §1 然后典范同态 (6)
ϕ : E ⊗ A F → E ⊗ A / J F
\phi : E \otimes_A F \to E \otimes_{A/\mathfrak{J}} F
ϕ : E ⊗ A F → E ⊗ A / J F *相应的到典范同态
ρ : A → A / J \rho : A \to A/\mathfrak{J} ρ : A → A / J 是身份。
引用了 1 · 被以下引用 1 引用了 命题 2
被以下引用 推论 3
对于所有α ˉ ∈ A / J \bar{\alpha} \in A/\mathfrak{J} α ˉ ∈ A / J x ∈ E x \in E x ∈ E y ∈ F y \in F y ∈ F x α ˉ = x α x \bar{\alpha} = x \alpha x α ˉ = xα α ˉ y = α y \bar{\alpha} y = \alpha y α ˉ y = α y α \alpha α ρ ( α ) = α ˉ \rho(\alpha) = \bar{\alpha} ρ ( α ) = α ˉ C = Z ( E × F ) C = \mathbf{Z}^{(E \times F)} C = Z ( E × F ) ( x α , y ) − ( x , α y ) (x \alpha, y) - (x, \alpha y) ( xα , y ) − ( x , α y ) C C C ( x α ˉ , y ) − ( x , α ˉ y ) (x \bar{\alpha}, y) - (x, \bar{\alpha} y) ( x α ˉ , y ) − ( x , α ˉ y )
根据命题 2 的假设和记号，令E ′ E' E ′ B B B F ′ F' F ′ B B B u : E ′ → E u : E' \to E u : E ′ → E ρ : B → A \rho : B \to A ρ : B → A u u u v v v B B B E ′ → ρ ∗ ( E ) E' \to \rho_*(E) E ′ → ρ ∗ ( E ) F ′ → ρ ∗ ( F ′ ) F' \to \rho_*(F') F ′ → ρ ∗ ( F ′ ) Z \mathbf{Z} Z w : E ′ ⊗ B F ′ → ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) w : E' \otimes_B F' \to \rho_*(E) \otimes_B \rho_*(F) w : E ′ ⊗ B F ′ → ρ ∗ ( E ) ⊗ B ρ ∗ ( F ) w ( x ′ ⊗ y ′ ) = u ( x ′ ) ⊗ v ( y ′ ) w(x' \otimes y') = u(x') \otimes v(y') w ( x ′ ⊗ y ′ ) = u ( x ′ ) ⊗ v ( y ′ ) x ′ ∈ E ′ , y ′ ∈ F ′ x' \in E',\ y' \in F' x ′ ∈ E ′ , y ′ ∈ F ′ Z \mathbf{Z} Z w ′ : E ′ ⊗ B F ′ → E ⊗ A F w' : E' \otimes_B F' \to E \otimes_A F w ′ : E ′ ⊗ B F ′ → E ⊗ A F w ′ ( x ′ ⊗ y ′ ) = u ( x ′ ) ⊗ v ( y ′ ) w'(x' \otimes y') = u(x') \otimes v(y') w ′ ( x ′ ⊗ y ′ ) = u ( x ′ ) ⊗ v ( y ′ ) x ′ ∈ E ′ , y ′ ∈ F ′ x' \in E',\ y' \in F' x ′ ∈ E ′ , y ′ ∈ F ′ u ⊗ v u \otimes v u ⊗ v ( u , v ) ↦ u ⊗ v (u, v) \mapsto u \otimes v ( u , v ) ↦ u ⊗ v Z \mathbf{Z} Z
Hom B ( E ′ , ρ ∗ ( E ) ) × Hom B ( F ′ , ρ ∗ ( F ) ) → Hom Z ( E ′ ⊗ B F ′ , E ⊗ A F ) .
\operatorname{Hom}_B(E', \rho_*(E)) \times \operatorname{Hom}_B(F', \rho_*(F)) \to \operatorname{Hom}_\mathbf{Z}(E' \otimes_B F', E \otimes_A F).
Hom B ( E ′ , ρ ∗ ( E )) × Hom B ( F ′ , ρ ∗ ( F )) → Hom Z ( E ′ ⊗ B F ′ , E ⊗ A F ) . 此外，如果C C C σ : C → B \sigma : C \to B σ : C → B E ′ ′ E'' E ′′ C C C F ′ ′ F'' F ′′ C C C u ′ : E ′ ′ → E ′ u' : E'' \to E' u ′ : E ′′ → E ′ v ′ : F ′ ′ → F ′ v' : F'' \to F' v ′ : F ′′ → F ′ σ \sigma σ
( u ∘ u ′ ) ⊗ ( v ∘ v ′ ) = ( u ⊗ v ) ∘ ( u ′ ⊗ v ′ ) .
(u \circ u') \otimes (v \circ v') = (u \otimes v) \circ (u' \otimes v').
( u ∘ u ′ ) ⊗ ( v ∘ v ′ ) = ( u ⊗ v ) ∘ ( u ′ ⊗ v ′ ) .
4. 张量积上的操作员； 张量积作为多模块
根据假设和no. 1的记号，对于A A A E E E F F F u u u v v v u ⊗ 1 F u \otimes 1_F u ⊗ 1 F 1 E ⊗ v 1_E \otimes v 1 E ⊗ v Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F u ↦ u ⊗ 1 F u \mapsto u \otimes 1_F u ↦ u ⊗ 1 F v ↦ 1 E ⊗ v v \mapsto 1_E \otimes v v ↦ 1 E ⊗ v 环同态
End A ( E ) → End Z ( E ⊗ A F )
\operatorname{End}_A(E) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F)
End A ( E ) → End Z ( E ⊗ A F ) （分别End A ( F ) → End Z ( E ⊗ A F ) \operatorname{End}_A(F) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F) End A ( F ) → End Z ( E ⊗ A F )
( u ⊗ 1 F ) ∘ ( 1 E ⊗ v ) = ( 1 E ⊗ v ) ∘ ( u ⊗ 1 F ) = u ⊗ v
(u \otimes 1_F) \circ (1_E \otimes v) = (1_E \otimes v) \circ (u \otimes 1_F) = u \otimes v
( u ⊗ 1 F ) ∘ ( 1 E ⊗ v ) = ( 1 E ⊗ v ) ∘ ( u ⊗ 1 F ) = u ⊗ v 因此 (§ 1, no. 14) E ⊗ A F E \otimes_A F E ⊗ A F End A ( E ) \operatorname{End}_A(E) End A ( E ) End A ( F ) \operatorname{End}_A(F) End A ( F ) left 双模 结构。
既然如此，假设在 E E E ( ( B i ′ ) ; A , ( C j ′ ) ) ((B'_i); A, (C'_j)) (( B i ′ ) ; A , ( C j ′ )) F F F ( A , ( B h ′ ′ ) ; ( C k c ′ ′ ) ) (A, (B''_h); (C''_{k_c})) ( A , ( B h ′′ ) ; ( C k c ′′ )) B i ′ → End A ( E ) B'_i \to \operatorname{End}_A(E) B i ′ → End A ( E ) C j ′ 0 → End A ( E ) C'_j{}^0 \to \operatorname{End}_A(E) C j ′ 0 → End A ( E ) B h ′ ′ → End A ( F ) B''_h \to \operatorname{End}_A(F) B h ′′ → End A ( F ) C k c ′ ′ 0 → End A ( F ) C''_{k_c}{}^0 \to \operatorname{End}_A(F) C k c ′′ 0 → End A ( F ) End A ( E ) → End Z ( E ⊗ A F ) \operatorname{End}_A(E) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F) End A ( E ) → End Z ( E ⊗ A F ) End A ( F ) → End Z ( E ⊗ A F ) \operatorname{End}_A(F) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F) End A ( F ) → End Z ( E ⊗ A F )
B i ′ → End Z ( E ⊗ A F ) , C j ′ 0 → End Z ( E ⊗ A F ) B h ′ ′ → End Z ( E ⊗ A F ) , C k c ′ ′ 0 → End Z ( E ⊗ A F )
\begin{align*}
B'_i &\to \operatorname{End}_\mathbf{Z}(E \otimes_A F), & C'_j{}^0 &\to \operatorname{End}_\mathbf{Z}(E \otimes_A F) \\
B''_h &\to \operatorname{End}_\mathbf{Z}(E \otimes_A F), & C''_{k_c}{}^0 &\to \operatorname{End}_\mathbf{Z}(E \otimes_A F)
\end{align*}
B i ′ B h ′′ → End Z ( E ⊗ A F ) , → End Z ( E ⊗ A F ) , C j ′ 0 C k c ′′ 0 → End Z ( E ⊗ A F ) → End Z ( E ⊗ A F ) 定义为 pairwise permutable 像；在其他字中，E ⊗ A F E \otimes_A F E ⊗ A F ( ( B i ′ ) , ( B h ′ ′ ) ; ( C j ′ ) , ( C k c ′ ′ ) ) ((B'_i), (B''_h); (C'_j), (C''_{k_c})) (( B i ′ ) , ( B h ′′ ) ; ( C j ′ ) , ( C k c ′′ )) A A A
((B'_i); A, (C'_j))-多模块 E 和 (A, (B''_h); (C''_k))-多模块 F。该多模块是普遍映射问题的解，类似于 no. 1 中考虑的；准确地说：
设 G 为 ((B'_i), (B''_h); (C'_j), (C''_k))-多模块。
引用了 1 · 被以下引用 2 引用了 命题 1
被以下引用 § 3. 。 张量积 , 学院
(a) 设 g 为多模 E ⊗ A \otimes_A ⊗ A × \times × f : ( x , y ) ↦ g ( x ⊗ y ) f : (x, y) \mapsto g(x \otimes y) f : ( x , y ) ↦ g ( x ⊗ y ) Z \mathbf{Z} Z
{ f ( μ i ′ x , y ) = μ i ′ f ( x , y ) , f ( x v j ′ , y ) = f ( x , y ) v j ′ f ( x , μ h ′ ′ y ) = μ h ′ ′ f ( x , y ) , f ( x , y v k ′ ′ ) = f ( x , y ) v k ′ ′
\begin{cases}
f(\mu'_i x, y) = \mu'_i f(x, y), & f(x v'_j, y) = f(x, y) v'_j \\
f(x, \mu''_h y) = \mu''_h f(x, y), & f(x, y v''_k) = f(x, y) v''_k
\end{cases}
{ f ( μ i ′ x , y ) = μ i ′ f ( x , y ) , f ( x , μ h ′′ y ) = μ h ′′ f ( x , y ) , f ( x v j ′ , y ) = f ( x , y ) v j ′ f ( x , y v k ′′ ) = f ( x , y ) v k ′′ 对于所有 x ∈ E , y ∈ F , μ i ′ ∈ B i ′ , v j ′ ∈ C j ′ , μ h ′ ′ ∈ B h ′ ′ , v k ′ ′ ∈ C k ′ ′ , i , j , h , k x \in E, y \in F, \mu'_i \in B'_i, v'_j \in C'_j, \mu''_h \in B''_h, v''_k \in C''_k, i, j, h, k x ∈ E , y ∈ F , μ i ′ ∈ B i ′ , v j ′ ∈ C j ′ , μ h ′′ ∈ B h ′′ , v k ′′ ∈ C k ′′ , i , j , h , k
(b) 相反，令 f 为 E × \times × Z \mathbf{Z} Z ⊗ A \otimes_A ⊗ A f ( x , y ) = g ( x ⊗ y ) f(x, y) = g(x \otimes y) f ( x , y ) = g ( x ⊗ y ) x ∈ E , y ∈ F x \in E, y \in F x ∈ E , y ∈ F
断言 (a) 紧随 E ⊗ A \otimes_A ⊗ A ( x ⊗ y ) v j ′ = ( x v j ′ ) ⊗ y (x \otimes y)v'_j = (xv'_j) \otimes y ( x ⊗ y ) v j ′ = ( x v j ′ ) ⊗ y Z \mathbf{Z} Z g ( x ⊗ y ) = f ( x , y ) g(x \otimes y) = f(x, y) g ( x ⊗ y ) = f ( x , y ) x ∈ E , y ∈ F x \in E, y \in F x ∈ E , y ∈ F x ⊗ y x \otimes y x ⊗ y Z \mathbf{Z} Z ⊗ A \otimes_A ⊗ A g ( μ ′ ( x ⊗ y ) ) = μ i ′ g ( x ⊗ y ) g(\mu'(x \otimes y)) = \mu'_i g(x \otimes y) g ( μ ′ ( x ⊗ y )) = μ i ′ g ( x ⊗ y ) g ( x ⊗ y ) = f ( x , y ) g(x \otimes y) = f(x, y) g ( x ⊗ y ) = f ( x , y )
如果E E E A A A E E E E ⊗ A ( s A d ) E \otimes_A (sA_d) E ⊗ A ( s A d ) h : x ↦ x ⊗ 1 h : x \mapsto x \otimes 1 h : x ↦ x ⊗ 1 A A A g g g g ( x ⊗ λ ) = x λ g(x \otimes \lambda) = x\lambda g ( x ⊗ λ ) = x λ x ∈ E , λ ∈ A x \in E, \lambda \in A x ∈ E , λ ∈ A
被以下引用 16 被以下引用 AC I § 2. 平模(*) , 命题 7, II § 11 , 注 , 推论 1 , 推论 3 , 推论 6 , 引理 3, II § 4 , 命题 4, II § 4 , 推论, II § 5 , 命题 2, II § 5 , 命题 6, II § 5 , 命题 26, II § 7 , III § 1. 。 代数 , 命题 3, III § 11 , 命题 13, III § 7 , 定义 1, VIII § 6
如果 g g g g ∘ h g \circ h g ∘ h 1 E 1_E 1 E h ∘ g h \circ g h ∘ g E ⊗ A ( s A d ) E \otimes_A (sA_d) E ⊗ A ( s A d ) x ⊗ y x \otimes y x ⊗ y Z \mathbf{Z} Z
我们通常会写E ⊗ A A E \otimes_A A E ⊗ A A E ⊗ A ( s A d ) E \otimes_A (sA_d) E ⊗ A ( s A d ) E ⊗ A A E \otimes_A A E ⊗ A A E E E E E E B B B A A A E E E E ⊗ A A E \otimes_A A E ⊗ A A B B B g g g h h h
现在让 F F F A A A ( s A d ) ⊗ A F (sA_d) \otimes_A F ( s A d ) ⊗ A F A ⊗ A F A \otimes_A F A ⊗ A F A A A A ⊗ A F A \otimes_A F A ⊗ A F F F F λ ⊗ x \lambda \otimes x λ ⊗ x λ x \lambda x λ x x ↦ 1 ⊗ x x \mapsto 1 \otimes x x ↦ 1 ⊗ x
特别地，存在( A , A ) (A, A) ( A , A ) ( s A d ) ⊗ A ( s A d ) (sA_d) \otimes_A (sA_d) ( s A d ) ⊗ A ( s A d ) s A d sA_d s A d λ ⊗ μ \lambda \otimes \mu λ ⊗ μ λ μ \lambda \mu λ μ
5. 张量积两个模超过交换环
令 C C C C C C E E E E E E E E E F F F C C C E ⊗ C F E \otimes_C F E ⊗ C F C C C γ ( x ⊗ y ) = ( γ x ) ⊗ y \gamma(x \otimes y) = (\gamma x) \otimes y γ ( x ⊗ y ) = ( γ x ) ⊗ y γ ( x ⊗ y ) = x ⊗ ( γ y ) \gamma(x \otimes y) = x \otimes (\gamma y) γ ( x ⊗ y ) = x ⊗ ( γ y ) ( γ x ) ⊗ y = x ⊗ ( γ y ) (\gamma x) \otimes y = x \otimes (\gamma y) ( γ x ) ⊗ y = x ⊗ ( γ y ) 相同 。从此以后，当我们将 E ⊗ C F E \otimes_C F E ⊗ C F C C C
σ : F ⊗ C E → E ⊗ C F
\sigma : F \otimes_C E \to E \otimes_C F
σ : F ⊗ C E → E ⊗ C F （no. 1，推论 2 到命题 1）则为C C C
由此定义可以得出，如果( a λ ) λ ∈ L (a_\lambda)_{\lambda \in L} ( a λ ) λ ∈ L ( b μ ) μ ∈ M (b_\mu)_{\mu \in M} ( b μ ) μ ∈ M C C C E E E F F F ( a λ ⊗ b μ ) (a_\lambda \otimes b_\mu) ( a λ ⊗ b μ ) C C C E ⊗ C F E \otimes_C F E ⊗ C F E E E F F F C C C E ⊗ C F E \otimes_C F E ⊗ C F
对于每个C C C G G G E × F E \times F E × F Z \mathbf{Z} Z f f f G G G
f ( γ x , y ) = f ( x , γ y ) = γ f ( x , y ) for x ∈ E , y ∈ F , γ ∈ C
f(\gamma x, y) = f(x, \gamma y) = \gamma f(x, y) \quad \text{for } x \in E, y \in F, \gamma \in C
f ( γ x , y ) = f ( x , γ y ) = γ f ( x , y ) for x ∈ E , y ∈ F , γ ∈ C 然后称为 C-双线性和形式 C-模，用 L 2 ( E , F ; G ) \mathscr{L}_2(E, F; G) L 2 ( E , F ; G )
L 2 ( E , F ; G ) → H o m C ( E ⊗ C F , G ) .
\mathscr{L}_2(E, F; G) \to \mathrm{Hom}_C(E \otimes_C F, G).
L 2 ( E , F ; G ) → Hom C ( E ⊗ C F , G ) . 令E'，F'为两个C-模和u : E → E ′ u : E \to E' u : E → E ′ v : F → F ′ v : F \to F' v : F → F ′ u ⊗ v u \otimes v u ⊗ v E ⊗ C F E \otimes_C F E ⊗ C F E ′ ⊗ C F ′ E' \otimes_C F' E ′ ⊗ C F ′ ( u , v ) ↦ u ⊗ v (u, v) \mapsto u \otimes v ( u , v ) ↦ u ⊗ v H o m C ( E , E ′ ) × H o m C ( F , F ′ ) \mathrm{Hom}_C(E, E') \times \mathrm{Hom}_C(F, F') Hom C ( E , E ′ ) × Hom C ( F , F ′ ) H o m C ( E ⊗ C F , E ′ ⊗ C F ′ ) \mathrm{Hom}_C(E \otimes_C F, E' \otimes_C F') Hom C ( E ⊗ C F , E ′ ⊗ C F ′ )
H o m C ( E , E ′ ) ⊗ C H o m C ( F , F ′ ) → H o m C ( E ⊗ C F , E ′ ⊗ C F ′ )
\mathrm{Hom}_C(E, E') \otimes_C \mathrm{Hom}_C(F, F') \to \mathrm{Hom}_C(E \otimes_C F, E' \otimes_C F')
Hom C ( E , E ′ ) ⊗ C Hom C ( F , F ′ ) → Hom C ( E ⊗ C F , E ′ ⊗ C F ′ ) 与张量积的每一个元素 u ⊗ v u \otimes v u ⊗ v
H o m C ( E , E ′ ) ⊗ C H o m C ( F , F ′ )
\mathrm{Hom}_C(E, E') \otimes_C \mathrm{Hom}_C(F, F')
Hom C ( E , E ′ ) ⊗ C Hom C ( F , F ′ ) 线性映射u ⊗ v u \otimes v u ⊗ v 不一定是单射或满射 (§ 4, 习题 2)。
6. E ⊗ A F E \otimes_A F E ⊗ A F
设* E、E'、E'' 为右 A-*模*、F 左 A-*模* 且
(14)
E ′ → u E → v E ′ ′ → 0
E' \xrightarrow{u} E \xrightarrow{v} E'' \to 0
E ′ u E v E ′′ → 0 正合序列线性映射.* 写作 u ˉ = u ⊗ 1 F , v ˉ = v ⊗ 1 F , \bar{u} = u \otimes 1_F, \bar{v} = v \otimes 1_F, u ˉ = u ⊗ 1 F , v ˉ = v ⊗ 1 F ,
被以下引用 8 被以下引用 命题 7, II § 11 , 推论 , 命题 6 , 习题 6, § 4 , 命题 14, II § 7 , VIII § 11. 。 格罗滕迪克群 , 命题 9, VIII § 11 , 命题 11, VIII § 6
(15)
E ′ ⊗ A F → u ˉ E ⊗ A F → v ˉ E ′ ′ ⊗ A F → 0
E' \otimes_A F \xrightarrow{\bar{u}} E \otimes_A F \xrightarrow{\bar{v}} E'' \otimes_A F \to 0
E ′ ⊗ A F u ˉ E ⊗ A F v ˉ E ′′ ⊗ A F → 0 的 Z \mathbf{Z} Z
凭借no. 2、公式(5)、v ˉ ∘ u ˉ = ( v ∘ u ) ⊗ 1 F = 0 \bar{v} \circ \bar{u} = (v \circ u) \otimes 1_F = 0 v ˉ ∘ u ˉ = ( v ∘ u ) ⊗ 1 F = 0 H = u ˉ ( E ′ ⊗ F ) H = \bar{u}(E' \otimes F) H = u ˉ ( E ′ ⊗ F ) L = Ker ( v ˉ ) L = \operatorname{Ker}(\bar{v}) L = Ker ( v ˉ ) v ˉ \bar{v} v ˉ u ˉ \bar{u} u ˉ M = ( E ⊗ F ) / H M = (E \otimes F)/H M = ( E ⊗ F ) / H Z \mathbf{Z} Z f f f E ′ ′ ⊗ F E'' \otimes F E ′′ ⊗ F f f f Z \mathbf{Z} Z g : E ′ ′ ⊗ F → M g : E'' \otimes F \to M g : E ′′ ⊗ F → M g ∘ f g \circ f g ∘ f f ∘ g f \circ g f ∘ g
设x ′ ′ ∈ E ′ ′ x'' \in E'' x ′′ ∈ E ′′ y ∈ F y \in F y ∈ F x ∈ E x \in E x ∈ E v ( x ) = x ′ ′ v(x) = x'' v ( x ) = x ′′ x 1 , x 2 x_1, x_2 x 1 , x 2 v ( x 1 ) = v ( x 2 ) = x ′ ′ v(x_1) = v(x_2) = x'' v ( x 1 ) = v ( x 2 ) = x ′′ ϕ : E ⊗ F → M \phi : E \otimes F \to M ϕ : E ⊗ F → M ϕ ( x 1 ⊗ y ) = ϕ ( x 2 ⊗ y ) \phi(x_1 \otimes y) = \phi(x_2 \otimes y) ϕ ( x 1 ⊗ y ) = ϕ ( x 2 ⊗ y ) v ( x ) = 0 v(x) = 0 v ( x ) = 0 ϕ ( x ⊗ y ) = 0 \phi(x \otimes y) = 0 ϕ ( x ⊗ y ) = 0 x = u ( x ′ ) x = u(x') x = u ( x ′ ) x ′ ∈ E ′ x' \in E' x ′ ∈ E ′ x ⊗ y = u ( x ′ ) ⊗ y = u ˉ ( x ′ ⊗ y ) ∈ H x \otimes y = u(x') \otimes y = \bar{u}(x' \otimes y) \in H x ⊗ y = u ( x ′ ) ⊗ y = u ˉ ( x ′ ⊗ y ) ∈ H x ∈ E x \in E x ∈ E ( x ′ ′ , y ) (x'', y) ( x ′′ , y ) ϕ ( x ⊗ y ) \phi(x \otimes y) ϕ ( x ⊗ y ) v ( x ) = x ′ ′ v(x) = x'' v ( x ) = x ′′ E ′ ′ × F E'' \times F E ′′ × F Z \mathbf{Z} Z v ( x λ ) = x ′ ′ λ v(x \lambda) = x'' \lambda v ( x λ ) = x ′′ λ ( x λ ) ⊗ y = x ⊗ ( λ y ) (x \lambda) \otimes y = x \otimes (\lambda y) ( x λ ) ⊗ y = x ⊗ ( λ y ) x ∈ E x \in E x ∈ E E ′ ′ ⊗ F E'' \otimes F E ′′ ⊗ F Z \mathbf{Z} Z g g g g ( x ′ ′ ⊗ y ) = ϕ ( x ⊗ y ) g(x'' \otimes y) = \phi(x \otimes y) g ( x ′′ ⊗ y ) = ϕ ( x ⊗ y ) y ∈ F , x ∈ E y \in F, x \in E y ∈ F , x ∈ E x ′ ′ = v ( x ) x'' = v(x) x ′′ = v ( x ) f ∘ g f \circ g f ∘ g
形式 x ′ ′ ⊗ y x'' \otimes y x ′′ ⊗ y E ′ ′ ⊗ F E'' \otimes F E ′′ ⊗ F f ∘ g f \circ g f ∘ g E ′ ′ ⊗ F E'' \otimes F E ′′ ⊗ F x ∈ E x \in E x ∈ E y ∈ F y \in F y ∈ F f ( ϕ ( x ⊗ y ) ) = v ( x ) ⊗ y f(\phi(x \otimes y)) = v(x) \otimes y f ( ϕ ( x ⊗ y )) = v ( x ) ⊗ y g ( f ( ϕ ( x ⊗ y ) ) ) = ϕ ( x ⊗ y ) g(f(\phi(x \otimes y))) = \phi(x \otimes y) g ( f ( ϕ ( x ⊗ y ))) = ϕ ( x ⊗ y ) ϕ ( x ⊗ y ) \phi(x \otimes y) ϕ ( x ⊗ y ) g ∘ f g \circ f g ∘ f
设 F、F'、F'' 为左A-模，E 右A-模且*
F ′ → s F → t F ′ ′ ⟶ 0
\begin{array}{ccccc}
F' & \xrightarrow{s} & F & \xrightarrow{t} & F'' \longrightarrow 0
\end{array}
F ′ s F t F ′′ ⟶ 0 (16) 正合序列 / 线性映射。写出 \bar{s} = l E \otimes s, \bar{t} = l_E \otimes t, \mathbf{Z} 同态* 的序列
E ⊗ A F ′ → s ˉ E ⊗ A F → t ˉ E ⊗ A F ′ ′ ⟶ 0
\begin{array}{cccccc}
E \otimes_A F' & \xrightarrow{\bar{s}} & E \otimes_A F & \xrightarrow{\bar{t}} & E \otimes_A F'' & \longrightarrow 0
\end{array}
E ⊗ A F ′ s ˉ E ⊗ A F t ˉ E ⊗ A F ′′ ⟶ 0 (17) *准确。
引用了 3 · 被以下引用 2 引用了 命题 1 , 推论 2 , 命题 5
被以下引用 引理 1, AC I § 2 , 命题 6
当 E（或 F）被视为左（或右）A^0 模时，F \otimes_{A^0} E 被识别为 E \otimes_A F，并且 F' \otimes_{A^0} E 和 F'' \otimes_{A^0} E 也有类似的识别（no. 1，推论 2 到命题 1）；推论立即从命题 5 得出。
给定两个正合序列 (14) 和 (16)，同态 v \otimes t : E \otimes_A F \to E'' \otimes_A F'' 是满射，其核等于*
Im ( u ⊗ 1 F ) + Im ( l E ⊗ s )
\operatorname{Im}(u \otimes 1_F) + \operatorname{Im}(l_E \otimes s)
Im ( u ⊗ 1 F ) + Im ( l E ⊗ s ) 引用了 2 · 被以下引用 4 引用了 命题 5 , 推论
被以下引用 推论 3, II § 11 , 注 , 习题 1, § 1 , 命题 3, III § 5
现在 v ⊗ t = ( v ⊗ 1 F ′ ′ ) ∘ ( 1 E ⊗ t ) v \otimes t = (v \otimes 1_{F''}) \circ (1_E \otimes t) v ⊗ t = ( v ⊗ 1 F ′′ ) ∘ ( 1 E ⊗ t ) v ⊗ t v \otimes t v ⊗ t z ∈ E ⊗ F z \in E \otimes F z ∈ E ⊗ F v ⊗ t v \otimes t v ⊗ t ( 1 E ⊗ t ) ( z ) (1_E \otimes t)(z) ( 1 E ⊗ t ) ( z ) v ⊗ 1 F ′ ′ v \otimes 1_{F''} v ⊗ 1 F ′′
u ⊗ 1 F ′ ′ : E ′ ⊗ F ′ ′ → E ⊗ F ′ ′ .
u \otimes 1_{F''}: E' \otimes F'' \to E \otimes F''.
u ⊗ 1 F ′′ : E ′ ⊗ F ′′ → E ⊗ F ′′ . 但由于同态 t : F → F ′ ′ t: F \to F'' t : F → F ′′
1 E ′ ⊗ t : E ′ ⊗ F → E ′ ⊗ F ′ ′
1_{E'} \otimes t: E' \otimes F \to E' \otimes F''
1 E ′ ⊗ t : E ′ ⊗ F → E ′ ⊗ F ′′ 由命题 5 的推论，因此 z z z a ∈ E ′ ⊗ F a \in E' \otimes F a ∈ E ′ ⊗ F
( l E ⊗ t ) ( z ) = ( u ⊗ t ) ( a ) .
(l_E \otimes t)(z) = (u \otimes t)(a).
( l E ⊗ t ) ( z ) = ( u ⊗ t ) ( a ) . 令 b = z − ( u ⊗ 1 F ) ( a ) b = z - (u \otimes 1_F)(a) b = z − ( u ⊗ 1 F ) ( a ) ( l E ⊗ t ) ( b ) = 0 (l_E \otimes t)(b) = 0 ( l E ⊗ t ) ( b ) = 0 l E ⊗ s l_E \otimes s l E ⊗ s b b b
其他字：
令E ′ E' E ′ A A A E E E A A A F F F F ′ F' F ′ Im ( E ′ ⊗ A F ) \operatorname{Im}(E' \otimes_A F) Im ( E ′ ⊗ A F ) Im ( E ⊗ A F ′ ) \operatorname{Im}(E \otimes_A F') Im ( E ⊗ A F ′ ) E ⊗ A F E \otimes_A F E ⊗ A F Z \mathbf{Z} Z E ′ ⊗ A F → E ⊗ A F E' \otimes_A F \to E \otimes_A F E ′ ⊗ A F → E ⊗ A F
E ⊗ A F ′ → E ⊗ A F .
E \otimes_A F' \to E \otimes_A F.
E ⊗ A F ′ → E ⊗ A F . 然后是一个规范的 Z \mathbf{Z} Z
被以下引用 2 被以下引用 命题 1, III § 4 , 命题 8, IV § 6
( 20 ) π : ( E / E ′ ) ⊗ A ( F / F ′ ) → ( E ⊗ A F ) / ( Im ( E ′ ⊗ A F ) + Im ( E ⊗ A F ′ ) ) (20)\quad \pi : (E/E') \otimes_A (F/F') \to (E \otimes_A F)/(\operatorname{Im}(E' \otimes_A F) + \operatorname{Im}(E \otimes_A F')) ( 20 ) π : ( E / E ′ ) ⊗ A ( F / F ′ ) → ( E ⊗ A F ) / ( Im ( E ′ ⊗ A F ) + Im ( E ⊗ A F ′ )) *这样，对于 ξ ∈ E / E ′ \xi \in E/E' ξ ∈ E / E ′ η ∈ F \eta \in F η ∈ F π ( ξ ⊗ η ) \pi(\xi \otimes \eta) π ( ξ ⊗ η ) x ⊗ y ∈ E ⊗ A F x \otimes y \in E \otimes_A F x ⊗ y ∈ E ⊗ A F x ∈ ξ x \in \xi x ∈ ξ y ∈ η y \in \eta y ∈ η
注，当E E E ( ( B i ′ ) ; A , ( C j ′ ) ) ((B'_i); A, (C'_j)) (( B i ′ ) ; A , ( C j ′ )) F F F ( A , ( B h ′ ′ ) ; ( C k ′ ′ ) ) (A, (B''_h); (C''_k)) ( A , ( B h ′′ ) ; ( C k ′′ )) E ′ E' E ′ F ′ F' F ′ E E E F F F ( ( B i ′ ) , ( B h ′ ′ ) ; ( C j ′ ) , ( C k ′ ′ ) ) ((B'_i), (B''_h); (C'_j), (C''_k)) (( B i ′ ) , ( B h ′′ ) ; ( C j ′ ) , ( C k ′′ ))
令a a a A A A F F F A A A a F aF a F λ x \lambda x λ x F F F Z \mathbf{Z} Z λ ∈ a \lambda \in a λ ∈ a x ∈ F x \in F x ∈ F Z \mathbf{Z} Z
被以下引用 1 被以下引用 命题 7, II § 11
( 21 ) π : ( A / a ) ⊗ A F → F / a F (21)\quad \pi : (A/a) \otimes_A F \to F/aF ( 21 ) π : ( A / a ) ⊗ A F → F / a F *这样，对于所有 λ ˉ ∈ A / a \bar{\lambda} \in A/a λ ˉ ∈ A / a x ∈ F x \in F x ∈ F π ( λ ˉ ⊗ x ) \pi(\bar{\lambda} \otimes x) π ( λ ˉ ⊗ x ) λ x \lambda x λ x a F aF a F λ ∈ λ ˉ \lambda \in \bar{\lambda} λ ∈ λ ˉ
特别是，对于A = Z A = \mathbf{Z} A = Z n n n Z \mathbf{Z} Z F F F ( Z / n Z ) ⊗ Z F (\mathbf{Z}/n\mathbf{Z}) \otimes_{\mathbf{Z}} F ( Z / n Z ) ⊗ Z F Z \mathbf{Z} Z F / n F F/nF F / n F
令A A A a a a A A A E E E F F F A A A a a a F F F ( A / a ) (A/a) ( A / a ) E ⊗ A F E \otimes_A F E ⊗ A F ( E / a E ) ⊗ A / a F (E/aE) \otimes_{A/a} F ( E / a E ) ⊗ A / a F
引用了 1 · 被以下引用 1 引用了 推论
被以下引用 命题 1, AC III § 5
F F F E ⊗ A F E \otimes_A F E ⊗ A F a a a ( A / a ) (A/a) ( A / a ) § 1 § 1 §1 E ′ = a E E' = aE E ′ = a E Im ( E ′ ⊗ A F ) = 0 \operatorname{Im}(E' \otimes_A F) = 0 Im ( E ′ ⊗ A F ) = 0 E ⊗ A F E \otimes_A F E ⊗ A F ( E / a E ) ⊗ A F (E/aE) \otimes_A F ( E / a E ) ⊗ A F ( E / a E ) ⊗ A / a F (E/aE) \otimes_{A/a} F ( E / a E ) ⊗ A / a F
令a , b a, b a , b C C C C C C ( C / a ) ⊗ C ( C / b ) (C/a) \otimes_C (C/b) ( C / a ) ⊗ C ( C / b ) C ( a + b ) C(a + b) C ( a + b )
被以下引用 1 被以下引用 注
7. 张量积 / 乘积和直和
令( E λ ) λ ∈ L (E_\lambda)_{\lambda \in L} ( E λ ) λ ∈ L A A A ( F μ ) μ ∈ M (F_\mu)_{\mu \in M} ( F μ ) μ ∈ M A A A C = ∏ λ ∈ L E λ , D = ∏ μ ∈ M F μ C = \prod_{\lambda \in L} E_\lambda, D = \prod_{\mu \in M} F_\mu C = ∏ λ ∈ L E λ , D = ∏ μ ∈ M F μ ( ( x λ ) , ( y μ ) ) ↦ ( x λ ⊗ y μ ) ((x_\lambda), (y_\mu)) \mapsto (x_\lambda \otimes y_\mu) (( x λ ) , ( y μ )) ↦ ( x λ ⊗ y μ ) C × D C \times D C × D Z \mathbf{Z} Z
∏ ( λ , μ ) ∈ L × M ( E λ ⊗ A F μ )
\prod_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu)
( λ , μ ) ∈ L × M ∏ ( E λ ⊗ A F μ ) Z \mathbf{Z} Z Z \mathbf{Z} Z f : ( ∏ λ ∈ L E λ ) ⊗ A ( ∏ μ ∈ M F μ ) → ∏ ( λ , μ ) ∈ L × M ( E λ ⊗ A F μ )
f : \left( \prod_{\lambda \in L} E_\lambda \right) \otimes_A \left( \prod_{\mu \in M} F_\mu \right) \to \prod_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu)
f : ( λ ∈ L ∏ E λ ) ⊗ A μ ∈ M ∏ F μ → ( λ , μ ) ∈ L × M ∏ ( E λ ⊗ A F μ ) f ( ( x λ ) ⊗ ( y μ ) ) = ( x λ ⊗ y μ ) f((x_\lambda) \otimes (y_\mu)) = (x_\lambda \otimes y_\mu) f (( x λ ) ⊗ ( y μ )) = ( x λ ⊗ y μ ) 当C = R L , D = S M , R C = R^L, D = S^M, R C = R L , D = S M , R S S S A A A u ⊗ v u \otimes v u ⊗ v u u u L L L R R R M M M v v v S S S L × M L \times M L × M ( λ , μ ) ↦ u ( λ ) ⊗ v ( μ ) (\lambda, \mu) \mapsto u(\lambda) \otimes v(\mu) ( λ , μ ) ↦ u ( λ ) ⊗ v ( μ ) R ⊗ A S R \otimes_A S R ⊗ A S
当E λ E_\lambda E λ ( ( B i ′ ) ; A , ( C j ′ ) ) ((B'_i); A, (C'_j)) (( B i ′ ) ; A , ( C j ′ )) F μ F_\mu F μ ( A , ( B h ′ ′ ) ; ( C k c ′ ′ ) ) (A, (B''_h); (C''_{k_c})) ( A , ( B h ′′ ) ; ( C k c ′′ )) ( ( B i ′ ) , ( B h ′ ′ ) ; ( C j ′ ) , ( C k c ′ ′ ) ) ((B'_i), (B''_h); (C'_j), (C''_{k_c})) (( B i ′ ) , ( B h ′′ ) ; ( C j ′ ) , ( C k c ′′ ))
现在考虑C C C D D D E = ⨁ λ ∈ L E λ E = \bigoplus_{\lambda \in L} E_\lambda E = ⨁ λ ∈ L E λ ⨁ μ ∈ M F μ \bigoplus_{\mu \in M} F_\mu ⨁ μ ∈ M F μ E → C , F → D E \to C, F \to D E → C , F → D Z \mathbf{Z} Z E ⊗ A F → C ⊗ A D E \otimes_A F \to C \otimes_A D E ⊗ A F → C ⊗ A D E ⊗ A F E \otimes_A F E ⊗ A F ∏ λ , μ ( E λ ⊗ A F μ ) \prod_{\lambda, \mu} (E_\lambda \otimes_A F_\mu) ∏ λ , μ ( E λ ⊗ A F μ ) Z \mathbf{Z} Z g g g
g ( ( x λ ) ⊗ ( y μ ) ) = ( x λ ⊗ y μ ) ;
g((x_\lambda) \otimes (y_\mu)) = (x_\lambda \otimes y_\mu);
g (( x λ ) ⊗ ( y μ )) = ( x λ ⊗ y μ ) ; 此外，由于家庭( x λ ) (x_\lambda) ( x λ ) ( y μ ) (y_\mu) ( y μ ) ( x λ ⊗ y μ ) (x_\lambda \otimes y_\mu) ( x λ ⊗ y μ ) g g g
g : ( ⨁ λ ∈ L E λ ) ⊗ A ( ⨁ μ ∈ M F μ ) → ⨁ ( λ , μ ) ∈ L × M ( E λ ⊗ A F μ ) ,
g : \left( \bigoplus_{\lambda \in L} E_\lambda \right) \otimes_A \left( \bigoplus_{\mu \in M} F_\mu \right) \to \bigoplus_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu),
g : ( λ ∈ L ⨁ E λ ) ⊗ A μ ∈ M ⨁ F μ → ( λ , μ ) ∈ L × M ⨁ ( E λ ⊗ A F μ ) , 这是与（22）相同的条件下的多模块同态。
典范映射 (23) 是双射的。
引用了 1 · 被以下引用 15 引用了 命题 6, II § 1
被以下引用 命题 2, AC I § 2 , II § 11. 。分级为模和环 , § 3. 。 张量积 , 注 2 , 推论 1 , 推论 3 , 推论 5 , 推论 6 , 引理 2, II § 4 , 引理 5, II § 4 , 推论, II § 5 , 命题 2, III § 4 , III § 5. 。 张量代数、张量 , 引理 2, VIII § 15 , 命题 5, VIII § 7
为了证明这一点，只需将直和G = ⨁ ( λ , μ ) ∈ L × M ( E λ ⊗ A F μ ) G = \bigoplus_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu) G = ⨁ ( λ , μ ) ∈ L × M ( E λ ⊗ A F μ ) Z \mathbf{Z} Z h h h E ⊗ A F E \otimes_A F E ⊗ A F g ∘ h g \circ h g ∘ h h ∘ g h \circ g h ∘ g G G G E ⊗ A F E \otimes_A F E ⊗ A F Z \mathbf{Z} Z Z \mathbf{Z} Z
h λ μ : E λ ⊗ A F μ → E ⊗ A F
h_{\lambda \mu} : E_\lambda \otimes_A F_\mu \to E \otimes_A F
h λ μ : E λ ⊗ A F μ → E ⊗ A F 对于每个有序对( λ , μ ) (\lambda, \mu) ( λ , μ ) h λ μ = i λ ⊗ j μ h_{\lambda \mu} = i_\lambda \otimes j_\mu h λ μ = i λ ⊗ j μ i λ : E λ → E i_\lambda : E_\lambda \to E i λ : E λ → E j μ : F μ → F j_\mu : F_\mu \to F j μ : F μ → F h ∘ g h \circ g h ∘ g ( ∑ λ x λ ) ⊗ ( ∑ μ y μ ) \left( \sum_\lambda x_\lambda \right) \otimes \left( \sum_\mu y_\mu \right) ( ∑ λ x λ ) ⊗ ( ∑ μ y μ ) Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F g ∘ h g \circ h g ∘ h ∑ λ , μ ( x λ ⊗ y μ ) \sum_{\lambda, \mu} (x_\lambda \otimes y_\mu) ∑ λ , μ ( x λ ⊗ y μ ) Z \mathbf{Z} Z G G G ( λ , μ ) (\lambda, \mu) ( λ , μ )
x λ ⊗ y μ ( x λ ∈ E λ , y μ ∈ F μ )
x_\lambda \otimes y_\mu \quad (x_\lambda \in E_\lambda, y_\mu \in F_\mu)
x λ ⊗ y μ ( x λ ∈ E λ , y μ ∈ F μ ) 生成 Z \mathbf{Z} Z E λ ⊗ A F μ E_\lambda \otimes_A F_\mu E λ ⊗ A F μ
令u λ : E λ → E λ ′ u_\lambda : E_\lambda \to E'_\lambda u λ : E λ → E λ ′ v μ : F μ → F μ ′ v_\mu : F_\mu \to F'_\mu v μ : F μ → F μ ′ A A A
( ⨁ λ E λ ) ⊗ A ( ⨁ μ F μ ) ⟶ ⨁ λ , μ ( E λ ⊗ A ( F μ ) ) ( ⨁ u λ ) ⊗ ( ⨁ v μ ) ↓ ↓ ( ⨁ ( u λ ⊗ v μ ) ) ( ⨁ λ E λ ′ ) ⊗ A ( ⨁ μ F μ ′ ) ⟶ ⨁ λ , μ ( E λ ′ ⊗ A F μ ′ )
\begin{array}{ccc}
\left( \bigoplus_\lambda E_\lambda \right) \otimes_A \left( \bigoplus_\mu F_\mu \right) & \longrightarrow & \bigoplus_{\lambda, \mu} (E_\lambda \otimes_A (F_\mu)) \\
(\bigoplus u_\lambda) \otimes (\bigoplus v_\mu) \downarrow & & \downarrow (\bigoplus (u_\lambda \otimes v_\mu)) \\
\left( \bigoplus_\lambda E'_\lambda \right) \otimes_A \left( \bigoplus_\mu F'_\mu \right) & \longrightarrow & \bigoplus_{\lambda, \mu} (E'_\lambda \otimes_A F'_\mu)
\end{array}
( ⨁ λ E λ ) ⊗ A ( ⨁ μ F μ ) ( ⨁ u λ ) ⊗ ( ⨁ v μ ) ↓ ( ⨁ λ E λ ′ ) ⊗ A ( ⨁ μ F μ ′ ) ⟶ ⟶ ⨁ λ , μ ( E λ ⊗ A ( F μ )) ↓ ( ⨁ ( u λ ⊗ v μ )) ⨁ λ , μ ( E λ ′ ⊗ A F μ ′ ) 是可交换的。
如果左A A A F F F ( b μ ) μ ∈ M (b_\mu)_{\mu \in M} ( b μ ) μ ∈ M E ⊗ A F E \otimes_A F E ⊗ A F ∑ μ ( x μ ⊗ b μ ) \sum_\mu (x_\mu \otimes b_\mu) ∑ μ ( x μ ⊗ b μ ) x μ ∈ E x_\mu \in E x μ ∈ E ( x μ ) (x_\mu) ( x μ ) Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F E ( M ) E^{(M)} E ( M ) Z \mathbf{Z} Z
引用了 2 · 被以下引用 9 引用了 命题 4 , 命题 7
被以下引用 引理 10, AC I § 2 , 推论 3 , 推论 4 , 引理 3, II § 4 , 命题 4, II § 5 , 推出, II § 7 , 命题 6, III § 4 , 命题 7, III § 4 , 命题 5, VIII § 5
The basis ( b μ ) (b_\mu) ( b μ ) F F F ⨁ μ ∈ M A b μ \bigoplus_{\mu \in M} Ab_\mu ⨁ μ ∈ M A b μ E ⊗ A F → ⨁ μ ∈ M ( E ⊗ A A b μ ) E \otimes_A F \to \bigoplus_{\mu \in M} (E \otimes_A Ab_\mu) E ⊗ A F → ⨁ μ ∈ M ( E ⊗ A A b μ )
如果 E 是 ((B'_i); A, (C'_j))-多模，典范同构 E ⊗_A F → E^{(M)} 是 ((B'_i); (C'_j))-多模同构。
特别是，如果 E 也承认基 (a_λ)_{λ ε L}，则每个 z ∈ E ⊗_A F 都可以以一种且仅一种方式写成形式 ∑ λ , μ ( a λ ξ λ μ ) ⊗ b μ \sum_{\lambda, \mu} (a_\lambda \xi_{\lambda\mu}) \otimes b_\mu ∑ λ , μ ( a λ ξ λ μ ) ⊗ b μ z ↦ ( ξ λ μ ) ( λ , μ ) ∈ L × M z \mapsto (\xi_{\lambda\mu})_{(\lambda, \mu) \in L \times M} z ↦ ( ξ λ μ ) ( λ , μ ) ∈ L × M
如果 E 和 F 是交换环 C 上的两个自由模，且 (a_λ)（分别为 (b_μ)）是 C 模块 E（分别为 F）的基，则 (a_λ ⊗ b_μ) 是 C 模块 E 的基 ⊗_C F.
被以下引用 9 被以下引用 II § 10. 。 矩阵 , § 3. 。 张量积 , 推论 7 , 注, II § 7 , III § 1. 。 代数 , 命题 4, III § 4 , 注, III § 5 , 定理 1, III § 7 , 命题 1, VIII § 12
滥用语言，基 (a_λ ⊗ b_μ) 是有时，称为基 (a_λ) 和 (b_μ) 的 张量积 。
注(1) 。设 E 为自由右A-模，F 为自由左A-模，E 的 (a_λ)_{λ ∈ L} 基和 F 的 (b_μ)_{μ ∈ M} 基。每个元素 z ∈ E ⊗_A F 可以唯一地写为 ∑ λ a λ ⊗ y λ \sum_\lambda a_\lambda \otimes y_\lambda ∑ λ a λ ⊗ y λ ∑ μ x μ ⊗ b μ \sum_\mu x_\mu \otimes b_\mu ∑ μ x μ ⊗ b μ y λ = ∑ μ η λ μ b μ y_\lambda = \sum_\mu \eta_{\lambda\mu} b_\mu y λ = ∑ μ η λ μ b μ x μ = ∑ λ a λ ξ λ μ x_\mu = \sum_\lambda a_\lambda \xi_{\lambda\mu} x μ = ∑ λ a λ ξ λ μ ξ λ μ = η λ μ \xi_{\lambda\mu} = \eta_{\lambda\mu} ξ λ μ = η λ μ
∑ λ ( a λ ⊗ ( ∑ μ η λ μ b μ ) ) = ∑ λ , μ ( ( a λ η λ μ ) ⊗ b μ ) = ∑ μ ( ( ∑ λ a λ η λ μ ) ⊗ b μ ) .
\sum_\lambda \left( a_\lambda \otimes \left( \sum_\mu \eta_{\lambda\mu} b_\mu \right) \right) = \sum_{\lambda, \mu} ((a_\lambda \eta_{\lambda\mu}) \otimes b_\mu) = \sum_\mu \left( \left( \sum_\lambda a_\lambda \eta_{\lambda\mu} \right) \otimes b_\mu \right).
λ ∑ ( a λ ⊗ ( μ ∑ η λ μ b μ ) ) = λ , μ ∑ (( a λ η λ μ ) ⊗ b μ ) = μ ∑ ( ( λ ∑ a λ η λ μ ) ⊗ b μ ) .
( ∏ λ ∈ L E λ ) ⊗ A F → ∏ λ ∈ L ( E λ ⊗ A F )
\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_A F \to \prod_{\lambda \in L} (E_\lambda \otimes_A F)
( λ ∈ L ∏ E λ ) ⊗ A F → λ ∈ L ∏ ( E λ ⊗ A F ) 是单射（或双射）。
如果(b_μ)是F的基，则( ∏ λ ∈ L E λ ) ⊗ A F \left( \prod_{\lambda \in L} E_\lambda \right) \otimes_A F ( ∏ λ ∈ L E λ ) ⊗ A F z = ∑ μ ( ( x λ ( μ ) ) ⊗ b μ ) z = \sum_\mu ((x_\lambda^{(\mu)}) \otimes b_\mu) z = ∑ μ (( x λ ( μ ) ) ⊗ b μ ) ∑ μ ( x λ ( μ ) ⊗ b μ ) = 0 \sum_\mu (x_\lambda^{(\mu)} \otimes b_\mu) = 0 ∑ μ ( x λ ( μ ) ⊗ b μ ) = 0 x λ ( μ ) = 0 x_\lambda^{(\mu)} = 0 x λ ( μ ) = 0 z = 0 z = 0 z = 0
证明当 F 承认有限基时典范映射是双射的，凭借命题 7，立即减少到以下情况：
F = A_s；但随后两侧都被规范地标识为∏ λ ∈ L E λ \prod_{\lambda \in L} E_\lambda ∏ λ ∈ L E λ
设 A 为环，没有除子或零，E 为自由右A-模，F 为自由左A-模。那么 E ⊗ A \otimes_A ⊗ A ⊗ \otimes ⊗
引用了 1 · 被以下引用 1 引用了 推论 1
被以下引用 命题 1, VII § 4
令( a λ ) (a_\lambda) ( a λ ) ( b μ ) (b_\mu) ( b μ ) x = ∑ λ a λ ξ λ , y = ∑ μ η μ b μ x = \sum_\lambda a_\lambda \xi_\lambda, y = \sum_\mu \eta_\mu b_\mu x = ∑ λ a λ ξ λ , y = ∑ μ η μ b μ x ⊗ y = ∑ λ , μ ( ( a λ ξ λ η μ ) ⊗ b μ x \otimes y = \sum_{\lambda, \mu} ((a_\lambda \xi_\lambda \eta_\mu) \otimes b_\mu x ⊗ y = ∑ λ , μ (( a λ ξ λ η μ ) ⊗ b μ x ⊗ y = 0 x \otimes y = 0 x ⊗ y = 0 ξ λ η μ = 0 \xi_\lambda \eta_\mu = 0 ξ λ η μ = 0 ( λ , μ ) (\lambda, \mu) ( λ , μ ) x ≠ 0 x \neq 0 x = 0 λ \lambda λ ξ λ ≠ 0 \xi_\lambda \neq 0 ξ λ = 0 η μ = 0 \eta_\mu = 0 η μ = 0 μ \mu μ y = 0 y = 0 y = 0
设 E 为 E 的右A-模、F 左A-模、M 子模和 F 的 N 子模。如果 M 为 E 的直因子且 F 的 N 直因子，则典范同态 M ⊗ A N → E ⊗ A F M \otimes_A N \to E \otimes_A F M ⊗ A N → E ⊗ A F M ⊗ A N M \otimes_A N M ⊗ A N Z \mathbf{Z} Z E ⊗ A F E \otimes_A F E ⊗ A F
引用了 1 · 被以下引用 4 引用了 命题 7
被以下引用 引理 1, AC I § 2 , 推论 7 , 命题 14, II § 7 , 命题 2, III § 11
这紧随命题 7 而来。
注，如果 E 是( ( B i ′ ) ; A , ( C j ′ ) ) ((B'_i); A, (C'_j)) (( B i ′ ) ; A , ( C j ′ )) ( A , ( B h ′ ′ ) ; ( C k ′ ′ ) ) (A, (B''_h); (C''_k)) ( A , ( B h ′′ ) ; ( C k ′′ )) M ⊗ N M \otimes N M ⊗ N ( ( B i ′ ) , ( B h ′ ′ ) ; ( C j ′ ) , ( C k ′ ′ ) ) ((B'_i), (B''_h); (C'_j), (C''_k)) (( B i ′ ) , ( B h ′′ ) ; ( C j ′ ) , ( C k ′′ )) E ⊗ F E \otimes F E ⊗ F
设 P 为射影左A-模，E、F 为射影右A-模。对于每个单同态 u : E → F u : E \to F u : E → F
u ⊗ 1 P : E ⊗ A P → F ⊗ A P
u \otimes 1_P : E \otimes_A P \to F \otimes_A P
u ⊗ 1 P : E ⊗ A P → F ⊗ A P *是单射的。
引用了 4 · 被以下引用 4 引用了 命题 4, II § 2 , 命题 7 , 推论 1, II § 1 , 命题 4
被以下引用 命题 7, II § 11 , 推论, II § 4 , 命题 9, VIII § 11 , 命题 11, VIII § 6
存在左A-模 Q，使得 L = P ⊕ Q L = P \oplus Q L = P ⊕ Q u ⊗ 1 L u \otimes 1_L u ⊗ 1 L
( u ⊗ 1 P ) ⊕ ( u ⊗ 1 Q ) ; (u \otimes 1_P) \oplus (u \otimes 1_Q); ( u ⊗ 1 P ) ⊕ ( u ⊗ 1 Q ) ; 因此，只要证明 P 是*自由*时的推论就足够了（§ 1, no. 6，推论 1 到命题 7）。与
P = A s P = A_s P = A s 的情况相同的参数约化问题，它紧随 no. 4、命题 4。
令C为交换环。如果 E 和 F 是两个射影 C 模，则 E ⊗ C F E \otimes_C F E ⊗ C F
引用了 2 引用了 推论 2 , 推论 5
这直接从推论 5 以及两个免费 C 模块中的张量积是一个免费 C 模块这一事实得出（推论 2）。
8. 结合性张量积
设 A、B 为两个环，E 右A-模，F 为 (A, B) 双模，G 左B-模。则 E ⊗_A F 为右B-模，F ⊗_B G 左A-模且存在且仅有一个 Z \mathbf{Z} Z
引用了 1 · 被以下引用 6 引用了 命题 1
被以下引用 命题 8, AC I § 2 , § 3. 。 张量积 , 命题 3, II § 5 , 命题 6, II § 5 , 例, III § 4 , 定义 1, VIII § 6
ϕ : ( E ⊗ A F ) ⊗ B G → E ⊗ A ( F ⊗ B G )
\phi : (E \otimes_A F) \otimes_B G \to E \otimes_A (F \otimes_B G)
ϕ : ( E ⊗ A F ) ⊗ B G → E ⊗ A ( F ⊗ B G ) 这样 ϕ ( ( x ⊗ y ) ⊗ z ) = x ⊗ ( y ⊗ z ) \phi((x \otimes y) \otimes z) = x \otimes (y \otimes z) ϕ (( x ⊗ y ) ⊗ z ) = x ⊗ ( y ⊗ z ) x ∈ E , y ∈ F , z ∈ G ; x \in E,\ y \in F,\ z \in G; x ∈ E , y ∈ F , z ∈ G ; Z \mathbf{Z} Z
E ⊗_A F 上的右B-模结构和 F ⊗_B G 上的左A-模结构已在no. 4 中定义。 ϕ \phi ϕ ( x ⊗ y ) ⊗ z (x \otimes y) \otimes z ( x ⊗ y ) ⊗ z Z \mathbf{Z} Z ( E ⊗ A F ) ⊗ B G (E \otimes_A F) \otimes_B G ( E ⊗ A F ) ⊗ B G ϕ \phi ϕ z ∈ G , h z : y ↦ y ⊗ z z \in G,\ h_z : y \mapsto y \otimes z z ∈ G , h z : y ↦ y ⊗ z g z = 1 E ⊗ h z g_z = 1_E \otimes h_z g z = 1 E ⊗ h z Z \mathbf{Z} Z ( t , z ) ↦ g z ( t ) (t, z) \mapsto g_z(t) ( t , z ) ↦ g z ( t ) ( E ⊗ A F × G (E \otimes_A F \times G ( E ⊗ A F × G E ⊗ A ( F ⊗ B G ) E \otimes_A (F \otimes_B G) E ⊗ A ( F ⊗ B G ) h z + z ′ = h z + h z ′ h_{z + z'} = h_z + h_{z'} h z + z ′ = h z + h z ′ z ∈ G , z ′ ∈ G z \in G,\ z' \in G z ∈ G , z ′ ∈ G Z \mathbf{Z} Z μ ∈ B , g μ z ( t ) = g z ( t μ ) \mu \in B,\ g_{\mu z}(t) = g_z(t \mu) μ ∈ B , g μ z ( t ) = g z ( t μ ) t = x ⊗ y t = x \otimes y t = x ⊗ y x ∈ E x \in E x ∈ E y ∈ F y \in F y ∈ F
g μ z ( x ⊗ y ) = x ⊗ ( y ⊗ μ z )
g_{\mu z}(x \otimes y) = x \otimes (y \otimes \mu z)
g μ z ( x ⊗ y ) = x ⊗ ( y ⊗ μ z ) g z ( ( x ⊗ y ) μ ) = g z ( x ⊗ y μ ) = x ⊗ ( y μ ⊗ z ) .
g_z((x \otimes y)\mu) = g_z(x \otimes y\mu) = x \otimes (y\mu \otimes z).
g z (( x ⊗ y ) μ ) = g z ( x ⊗ y μ ) = x ⊗ ( y μ ⊗ z ) . 命题 1 (no. 1) 然后证明Z \mathbf{Z} Z
ϕ : ( E ⊗ A F ) ⊗ B G → E ⊗ A ( F ⊗ B G )
\phi : (E \otimes_A F) \otimes_B G \to E \otimes_A (F \otimes_B G)
ϕ : ( E ⊗ A F ) ⊗ B G → E ⊗ A ( F ⊗ B G ) 这样ϕ ( t ⊗ z ) = g z ( t ) \phi(t \otimes z) = g_z(t) ϕ ( t ⊗ z ) = g z ( t ) ϕ ( ( x ⊗ y ) ⊗ z ) = x ⊗ ( y ⊗ z ) \phi((x \otimes y) \otimes z) = x \otimes (y \otimes z) ϕ (( x ⊗ y ) ⊗ z ) = x ⊗ ( y ⊗ z ) Z \mathbf{Z} Z
ψ : E ⊗ A ( F ⊗ B G ) → ( E ⊗ A F ) ⊗ B G
\psi : E \otimes_A (F \otimes_B G) \to (E \otimes_A F) \otimes_B G
ψ : E ⊗ A ( F ⊗ B G ) → ( E ⊗ A F ) ⊗ B G 被定义为使得ψ ( x ⊗ ( y ⊗ z ) ) = ( x ⊗ y ) ⊗ z \psi(x \otimes (y \otimes z)) = (x \otimes y) \otimes z ψ ( x ⊗ ( y ⊗ z )) = ( x ⊗ y ) ⊗ z ψ ∘ ϕ \psi \circ \phi ψ ∘ ϕ ϕ ∘ ψ \phi \circ \psi ϕ ∘ ψ ( E ⊗ A F ) ⊗ B G (E \otimes_A F) \otimes_B G ( E ⊗ A F ) ⊗ B G E ⊗ A ( F ⊗ B G ) E \otimes_A (F \otimes_B G) E ⊗ A ( F ⊗ B G ) Z \mathbf{Z} Z
直接的，如果E E E ( ( C i ′ ) ; A , ( D j ′ ) ) ((C'_i); A, (D'_j)) (( C i ′ ) ; A , ( D j ′ )) F F F ( A , ( C h ′ ′ ) ; B , ( D k ′ ′ ) ) (A, (C''_h); B, (D''_k)) ( A , ( C h ′′ ) ; B , ( D k ′′ )) G G G ( B , ( C l ′ ′ ′ ) ; ( D m ′ ′ ′ ) ) (B, (C'''_l); (D'''_m)) ( B , ( C l ′′′ ) ; ( D m ′′′ )) ( ( C i ′ ) , ( C h ′ ′ ) , ( C l ′ ′ ′ ) ; ( D j ′ ) , ( D k ′ ′ ) , ( D m ′ ′ ′ ) ) ((C'_i), (C''_h), (C'''_l); (D'_j), (D''_k), (D'''_m)) (( C i ′ ) , ( C h ′′ ) , ( C l ′′′ ) ; ( D j ′ ) , ( D k ′′ ) , ( D m ′′′ )) C C C E , F , G E, F, G E , F , G C C C C C C
( E ⊗ C F ) ⊗ C G → E ⊗ C ( F ⊗ C G ) .
(E \otimes_C F) \otimes_C G \to E \otimes_C (F \otimes_C G).
( E ⊗ C F ) ⊗ C G → E ⊗ C ( F ⊗ C G ) . 我们将在下面看到，在某些条件下，张量积的定义可以推广到一系列多模块，这将特别在命题8的假设下为我们提供一个Z \mathbf{Z} Z E ⊗ A F ⊗ B G E \otimes_A F \otimes_B G E ⊗ A F ⊗ B G Z \mathbf{Z} Z ( E ⊗ A F ) ⊗ B G (E \otimes_A F) \otimes_B G ( E ⊗ A F ) ⊗ B G E ⊗ A ( F ⊗ B G ) E \otimes_A (F \otimes_B G) E ⊗ A ( F ⊗ B G )
9. 多模块系列的张量积
令( G λ ) λ ∈ L (G_\lambda)_{\lambda \in L} ( G λ ) λ ∈ L Z \mathbf{Z} Z ( x λ ) ↦ u ( ( x λ ) ) (x_\lambda) \mapsto u((x_\lambda)) ( x λ ) ↦ u (( x λ )) x λ x_\lambda x λ G = ∏ λ ∈ L G λ G = \prod_{\lambda \in L} G_\lambda G = ∏ λ ∈ L G λ u u u Z \mathbf{Z} Z Z \mathbf{Z} Z μ ∈ L \mu \in L μ ∈ L ( a λ ) ∈ ∏ λ ≠ μ G λ (a_\lambda) \in \prod_{\lambda \neq \mu} G_\lambda ( a λ ) ∈ ∏ λ = μ G λ G G G G μ × ∏ λ ≠ μ G λ G_\mu \times \prod_{\lambda \neq \mu} G_\lambda G μ × ∏ λ = μ G λ
u ( x μ + y μ , ( a λ ) ) = u ( x μ , ( a λ ) ) + u ( y μ , ( a λ ) ) for x μ , y μ in G μ .
u(x_\mu + y_\mu, (a_\lambda)) = u(x_\mu, (a_\lambda)) + u(y_\mu, (a_\lambda)) \quad \text{for } x_\mu, y_\mu \text{ in } G_\mu.
u ( x μ + y μ , ( a λ )) = u ( x μ , ( a λ )) + u ( y μ , ( a λ )) for x μ , y μ in G μ . 这特别意味着如果x λ x_\lambda x λ u ( ( x λ ) ) = 0 u((x_\lambda)) = 0 u (( x λ )) = 0
我们还考虑*普遍映射问题*，其中Σ \Sigma Σ Z \mathbf{Z} Z α \alpha α G G G Z \mathbf{Z} Z G G G Z \mathbf{Z} Z C = Z ( G ) C = \mathbf{Z}^{(G)} C = Z ( G ) Z \mathbf{Z} Z C C C Z \mathbf{Z} Z D D D
( x μ + y μ , ( z λ ) λ ≠ μ ) − ( x μ , ( z λ ) λ ≠ μ ) − ( y μ , ( z λ ) λ ≠ μ )
(x_\mu + y_\mu, (z_\lambda)_{\lambda \neq \mu}) - (x_\mu, (z_\lambda)_{\lambda \neq \mu}) - (y_\mu, (z_\lambda)_{\lambda \neq \mu})
( x μ + y μ , ( z λ ) λ = μ ) − ( x μ , ( z λ ) λ = μ ) − ( y μ , ( z λ ) λ = μ ) 其中 μ ∈ L , x μ ∈ G μ , y μ ∈ G μ \mu \in L, x_\mu \in G_\mu, y_\mu \in G_\mu μ ∈ L , x μ ∈ G μ , y μ ∈ G μ z λ ∈ G λ z_\lambda \in G_\lambda z λ ∈ G λ λ ≠ μ \lambda \neq \mu λ = μ Z \mathbf{Z} Z C / D C/D C / D Z \mathbf{Z} Z ( G λ ) λ ∈ L (G_\lambda)_{\lambda \in L} ( G λ ) λ ∈ L Z \mathbf{Z} Z ⨂ λ ∈ L G λ \bigotimes_{\lambda \in L} G_\lambda ⨂ λ ∈ L G λ G G G ( x λ ) λ ∈ L (x_\lambda)_{\lambda \in L} ( x λ ) λ ∈ L C C C ⨂ λ ∈ L x λ \bigotimes_{\lambda \in L} x_\lambda ⨂ λ ∈ L x λ ⨂ λ ∈ L G λ \bigotimes_{\lambda \in L} G_\lambda ⨂ λ ∈ L G λ ϕ : ( x λ ) ↦ ⨂ λ ∈ L x λ \phi : (x_\lambda) \mapsto \bigotimes_{\lambda \in L} x_\lambda ϕ : ( x λ ) ↦ ⨂ λ ∈ L x λ Z \mathbf{Z} Z Z \mathbf{Z} Z Z \mathbf{Z} Z f f f Z \mathbf{Z} Z g : ⨂ λ ∈ L G λ → H g : \bigotimes_{\lambda \in L} G_\lambda \to H g : ⨂ λ ∈ L G λ → H f = g ∘ ϕ f = g \circ \phi f = g ∘ ϕ ( ⨂ λ ∈ L G λ , ϕ ) \left( \bigotimes_{\lambda \in L} G_\lambda, \phi \right) ( ⨂ λ ∈ L G λ , ϕ )
令( G λ ′ ) λ ∈ L (G'_\lambda)_{\lambda \in L} ( G λ ′ ) λ ∈ L Z \mathbf{Z} Z λ ∈ L \lambda \in L λ ∈ L v λ : G λ → G λ ′ v_\lambda : G_\lambda \to G'_\lambda v λ : G λ → G λ ′ Z \mathbf{Z} Z
( x λ ) ↦ ⨂ λ ∈ L v λ ( x λ )
(x_\lambda) \mapsto \bigotimes_{\lambda \in L} v_\lambda(x_\lambda)
( x λ ) ↦ λ ∈ L ⨂ v λ ( x λ ) ⨂ λ ∈ L G λ ′ \bigotimes_{\lambda \in L} G'_\lambda ⨂ λ ∈ L G λ ′ Z \mathbf{Z} Z ⨂ λ ∈ L G λ \bigotimes_{\lambda \in L} G_\lambda ⨂ λ ∈ L G λ ⨂ λ ∈ L G λ ′ \bigotimes_{\lambda \in L} G'_\lambda ⨂ λ ∈ L G λ ′ Z \mathbf{Z} Z ⨂ λ ∈ L v λ \bigotimes_{\lambda \in L} v_\lambda ⨂ λ ∈ L v λ ( ⨂ λ ∈ L v λ ) ( ⨂ λ ∈ L x λ ) = ⨂ λ ∈ L v λ ( x λ ) .
\left( \bigotimes_{\lambda \in L} v_\lambda \right) \left( \bigotimes_{\lambda \in L} x_\lambda \right) = \bigotimes_{\lambda \in L} v_\lambda(x_\lambda).
( λ ∈ L ⨂ v λ ) ( λ ∈ L ⨂ x λ ) = λ ∈ L ⨂ v λ ( x λ ) . μ ∈ L \mu \in L μ ∈ L θ \theta θ G μ G_\mu G μ ⨂ λ ∈ L G λ \bigotimes_{\lambda \in L} G_\lambda ⨂ λ ∈ L G λ θ ~ \tilde{\theta} θ ~ ⨂ λ ∈ L v λ \bigotimes_{\lambda \in L} v_\lambda ⨂ λ ∈ L v λ v μ = θ v_\mu = \theta v μ = θ v λ = 1 G λ v_\lambda = 1_{G_\lambda} v λ = 1 G λ λ ≠ μ \lambda \neq \mu λ = μ 然后我们假设给定集合 Ω \Omega Ω
c : ω ↦ ( ρ ( ω ) , σ ( ω ) )
c : \omega \mapsto (\rho(\omega), \sigma(\omega))
c : ω ↦ ( ρ ( ω ) , σ ( ω )) Ω \Omega Ω L × L L \times L L × L ω ∈ Ω \omega \in \Omega ω ∈ Ω G ρ ( ω ) G_{\rho(\omega)} G ρ ( ω ) p ω p_\omega p ω G σ ( ω ) G_{\sigma(\omega)} G σ ( ω ) q ω q_\omega q ω p ~ ω \tilde{p}_\omega p ~ ω q ~ ω \tilde{q}_\omega q ~ ω P = ⨂ λ ∈ L G λ P = \bigotimes_{\lambda \in L} G_\lambda P = ⨂ λ ∈ L G λ ω \omega ω Ω \Omega Ω p ~ ω − q ~ ω \tilde{p}_\omega - \tilde{q}_\omega p ~ ω − q ~ ω Z \mathbf{Z} Z Z \mathbf{Z} Z P / R P/R P / R ( G λ ) λ ∈ L (G_\lambda)_{\lambda \in L} ( G λ ) λ ∈ L ⨂ ( c , p , q ) G λ \bigotimes_{(c, p, q)} G_\lambda ⨂ ( c , p , q ) G λ P → P / R P \to P/R P → P / R ϕ : G → ⨂ λ ∈ L G λ \phi : G \to \bigotimes_{\lambda \in L} G_\lambda ϕ : G → ⨂ λ ∈ L G λ Z \mathbf{Z} Z ϕ ( c , p , q ) : G → ⨂ ( c , p , q ) G λ \phi_{(c, p, q)} : G \to \bigotimes_{(c, p, q)} G_\lambda ϕ ( c , p , q ) : G → ⨂ ( c , p , q ) G λ ϕ ( c , p , q ) ( ( x λ ) ) = ⨂ ( c , p , q ) x λ \phi_{(c, p, q)}((x_\lambda)) = \bigotimes_{(c, p, q)} x_\lambda ϕ ( c , p , q ) (( x λ )) = ⨂ ( c , p , q ) x λ ⨂ ( c ) x λ \bigotimes_{(c)} x_\lambda ⨂ ( c ) x λ ⨂ ( c , p , q ) x λ \bigotimes_{(c, p, q)} x_\lambda ⨂ ( c , p , q ) x λ ϕ ( c , p , q ) \phi_{(c, p, q)} ϕ ( c , p , q ) 普遍映射问题 ：令 p ˉ ω \bar{p}_\omega p ˉ ω q ˉ ω \bar{q}_\omega q ˉ ω ( x ρ ( ω ) , ( x λ ) λ ≠ ρ ( ω ) ) ↦ ( p ω ( x ρ ( ω ) ) , ( x λ ) λ ≠ ρ ( ω ) )
(x_{\rho(\omega)}, (x_\lambda)_{\lambda \neq \rho(\omega)}) \mapsto (p_\omega(x_{\rho(\omega)}), (x_\lambda)_{\lambda \neq \rho(\omega)})
( x ρ ( ω ) , ( x λ ) λ = ρ ( ω ) ) ↦ ( p ω ( x ρ ( ω ) ) , ( x λ ) λ = ρ ( ω ) ) ( x σ ( ω ) , ( x λ ) λ ≠ σ ( ω ) ) ↦ ( q ω ( x σ ( ω ) ) , ( x λ ) λ ≠ σ ( ω ) ) (x_{\sigma(\omega)}, (x_\lambda)_{\lambda \neq \sigma(\omega)}) \mapsto (q_\omega(x_{\sigma(\omega)}), (x_\lambda)_{\lambda \neq \sigma(\omega)}) ( x σ ( ω ) , ( x λ ) λ = σ ( ω ) ) ↦ ( q ω ( x σ ( ω ) ) , ( x λ ) λ = σ ( ω ) ) Σ \Sigma Σ Z \mathbf{Z} Z G G G Z \mathbf{Z} Z u u u Z \mathbf{Z} Z u ∘ p ˉ ω = u ∘ q ˉ ω
u \circ \bar{p}_\omega = u \circ \bar{q}_\omega
u ∘ p ˉ ω = u ∘ q ˉ ω 对于所有ω ∈ Ω \omega \in \Omega ω ∈ Ω
这个构造特别恢复了no. 1中描述的E ⊗ A F E \otimes_A F E ⊗ A F L = { 1 , 2 } L = \{1, 2\} L = { 1 , 2 } G 1 = E G_1 = E G 1 = E G 2 = F G_2 = F G 2 = F Ω = A \Omega = A Ω = A ω ∈ A \omega \in A ω ∈ A ρ ( ω ) = 1 \rho(\omega) = 1 ρ ( ω ) = 1 σ ( ω ) = 2 \sigma(\omega) = 2 σ ( ω ) = 2 ω \omega ω Z \mathbf{Z} Z E E E x ↦ x ω x \mapsto x\omega x ↦ x ω q ω q_\omega q ω y ↦ ω y y \mapsto \omega y y ↦ ω y Z \mathbf{Z} Z F F F
令( G λ ′ ) λ ∈ L (G'_\lambda)_{\lambda \in L} ( G λ ′ ) λ ∈ L Z \mathbf{Z} Z c c c G ρ ( ω ) ′ G'_{\rho(\omega)} G ρ ( ω ) ′ ω ∈ Ω \omega \in \Omega ω ∈ Ω p ω ′ p'_\omega p ω ′ G σ ( ω ) ′ G'_{\sigma(\omega)} G σ ( ω ) ′ q ω ′ q'_\omega q ω ′ λ ∈ L \lambda \in L λ ∈ L v λ : G λ → G λ ′ v_\lambda : G_\lambda \to G'_\lambda v λ : G λ → G λ ′ Z \mathbf{Z} Z ω ∈ Ω \omega \in \Omega ω ∈ Ω
v ρ ( ω ) ∘ p ω = p ω ′ ∘ v ρ ( ω ) and v σ ( ω ) ∘ q ω = q ω ′ ∘ v σ ( ω )
v_{\rho(\omega)} \circ p_\omega = p'_\omega \circ v_{\rho(\omega)} \quad \text{and} \quad v_{\sigma(\omega)} \circ q_\omega = q'_\omega \circ v_{\sigma(\omega)}
v ρ ( ω ) ∘ p ω = p ω ′ ∘ v ρ ( ω ) and v σ ( ω ) ∘ q ω = q ω ′ ∘ v σ ( ω ) （在其他字中，对于所有λ ∈ L \lambda \in L λ ∈ L v λ v_\lambda v λ G λ G_\lambda G λ G λ ′ G'_\lambda G λ ′ p ξ p_\xi p ξ q η q_\eta q η p ξ ′ p'_\xi p ξ ′ q η ′ q'_\eta q η ′ ξ \xi ξ η \eta η ρ ( ξ ) = λ \rho(\xi) = \lambda ρ ( ξ ) = λ σ ( η ) = λ \sigma(\eta) = \lambda σ ( η ) = λ
u : ( x λ ) ↦ ⨂ ( c , p ′ , q ′ ) v λ ( x λ )
u : (x_\lambda) \mapsto \bigotimes_{(c, p', q')} v_\lambda(x_\lambda)
u : ( x λ ) ↦ ( c , p ′ , q ′ ) ⨂ v λ ( x λ ) G G G ⨂ ( c , p ′ , q ′ ) G λ ′ \bigotimes_{(c, p', q')} G'_\lambda ⨂ ( c , p ′ , q ′ ) G λ ′ Z \mathbf{Z} Z ⨂ ( c , p , q ) G λ \bigotimes_{(c, p, q)} G_\lambda ⨂ ( c , p , q ) G λ ⨂ ( c , p ′ , q ′ ) G λ ′ \bigotimes_{(c, p', q')} G'_\lambda ⨂ ( c , p ′ , q ′ ) G λ ′ Z \mathbf{Z} Z ⨂ ( c ) v λ \bigotimes_{(c)} v_\lambda ⨂ ( c ) v λ 现在我们将为如此定义的一般张量积给出“结合性”性质。令( L i ) 1 ≤ i ≤ n (L_i)_{1 \leq i \leq n} ( L i ) 1 ≤ i ≤ n L L L i i i Ω i \Omega_i Ω i Ω \Omega Ω ρ ( ω ) ∈ L i \rho(\omega) \in L_i ρ ( ω ) ∈ L i σ ( ω ) ∈ L i \sigma(\omega) \in L_i σ ( ω ) ∈ L i Ω i \Omega_i Ω i Ω ′ = Ω − ( ⋃ i Ω i ) \Omega' = \Omega - \left( \bigcup_i \Omega_i \right) Ω ′ = Ω − ( ⋃ i Ω i ) i i i c ( i ) c^{(i)} c ( i ) Ω i \Omega_i Ω i ω ↦ ( ρ ( ω ) , σ ( ω ) ) \omega \mapsto (\rho(\omega), \sigma(\omega)) ω ↦ ( ρ ( ω ) , σ ( ω )) L i × L i L_i \times L_i L i × L i ω ∈ Ω i \omega \in \Omega_i ω ∈ Ω i p ω ( i ) p^{(i)}_\omega p ω ( i ) q ω ( i ) q^{(i)}_\omega q ω ( i ) p ω p_\omega p ω q ω q_\omega q ω i i i
F i = ⨂ ( c ( i ) , p ( i ) , q ( i ) ) G λ .
F_i = \bigotimes_{(c^{(i)}, p^{(i)}, q^{(i)})} G_\lambda.
F i = ( c ( i ) , p ( i ) , q ( i ) ) ⨂ G λ . 我们进一步做出以下“可置换性”假设：
(P) 如果ω ∈ Ω ′ \omega \in \Omega' ω ∈ Ω ′ p ω p_\omega p ω q ω q_\omega q ω G ρ ( ω ) G_{\rho(\omega)} G ρ ( ω ) G σ ( ω ) G_{\sigma(\omega)} G σ ( ω ) p ξ p_\xi p ξ q η q_\eta q η ξ ∉ Ω ′ \xi \notin \Omega' ξ ∈ / Ω ′ η ∉ Ω ′ \eta \notin \Omega' η ∈ / Ω ′ ρ ( ω ) = ρ ( ξ ) = σ ( η ) \rho(\omega) = \rho(\xi) = \sigma(\eta) ρ ( ω ) = ρ ( ξ ) = σ ( η ) σ ( ω ) = ρ ( ξ ) = σ ( η ) \sigma(\omega) = \rho(\xi) = \sigma(\eta) σ ( ω ) = ρ ( ξ ) = σ ( η )
对于每个ω ∈ Ω ′ \omega \in \Omega' ω ∈ Ω ′ i i i ρ ( ω ) ∈ L i \rho(\omega) \in L_i ρ ( ω ) ∈ L i ( v λ ) λ ∈ L i (v_\lambda)_{\lambda \in L_i} ( v λ ) λ ∈ L i v ρ ( ω ) = p ω v_{\rho(\omega)} = p_\omega v ρ ( ω ) = p ω v λ = 1 G λ v_\lambda = 1_{G_\lambda} v λ = 1 G λ λ ≠ ρ ( ω ) \lambda \neq \rho(\omega) λ = ρ ( ω ) ( v λ ) (v_\lambda) ( v λ ) p ′ p' p ′ p p p p ( i ) p^{(i)} p ( i ) q ′ q' q ′ q q q q ( i ) q^{(i)} q ( i ) ω \omega ω ξ \xi ξ Ω i \Omega_i Ω i ⨂ ( c ( i ) ) v λ = r ω \bigotimes_{(c^{(i)})} v_\lambda = r_\omega ⨂ ( c ( i ) ) v λ = r ω Z \mathbf{Z} Z F i F_i F i s ω s_\omega s ω q ω q_\omega q ω Z \mathbf{Z} Z F j F_j F j j j j σ ( ω ) ∈ L j \sigma(\omega) \in L_j σ ( ω ) ∈ L j d ( ω ) = ( i , j ) d(\omega) = (i, j) d ( ω ) = ( i , j ) 张量积 ⨂ ( d , r , s ) F i \bigotimes_{(d, r, s)} F_i ⨂ ( d , r , s ) F i
ϕ ( d , r , s ) : ∏ i = 1 n F i → ⨂ ( d , r , s ) F i .
\phi_{(d, r, s)} : \prod_{i=1}^n F_i \to \bigotimes_{(d, r, s)} F_i.
ϕ ( d , r , s ) : i = 1 ∏ n F i → ( d , r , s ) ⨂ F i . 另一方面，对于每个i i i
ψ i = ϕ ( c ( i ) , p ( i ) , q ( i ) ) : ∏ λ ∈ L i G λ → F i ;
\psi_i = \phi_{(c^{(i)}, p^{(i)}, q^{(i)})} : \prod_{\lambda \in L_i} G_\lambda \to F_i;
ψ i = ϕ ( c ( i ) , p ( i ) , q ( i ) ) : λ ∈ L i ∏ G λ → F i ; 使用集合的乘积的结合性，导出G G G ⨂ ( d , r , s ) F i \bigotimes_{(d, r, s)} F_i ⨂ ( d , r , s ) F i Z \mathbf{Z} Z ψ = ϕ ( d , r , s ) ∘ ( ψ i ) \psi = \phi_{(d, r, s)} \circ (\psi_i) ψ = ϕ ( d , r , s ) ∘ ( ψ i ) ( ⨂ ( d , r , s ) F i , ψ ) \left( \bigotimes_{(d, r, s)} F_i, \psi \right) ( ⨂ ( d , r , s ) F i , ψ ) 是相同通用问题的解 ( ⨂ ( c , p , q ) G λ , ϕ ( c , p , q ) ) \left( \bigotimes_{(c, p, q)} G_\lambda, \phi_{(c, p, q)} \right) ( ⨂ ( c , p , q ) G λ , ϕ ( c , p , q ) ) Z \mathbf{Z} Z
θ : ⨂ ( c , p , q ) G λ → ⨂ ( d , r , s ) F i
\theta : \bigotimes_{(c, p, q)} G_\lambda \to \bigotimes_{(d, r, s)} F_i
θ : ( c , p , q ) ⨂ G λ → ( d , r , s ) ⨂ F i 这样ψ = θ ∘ ϕ ( c , p , q ) \psi = \theta \circ \phi_{(c, p, q)} ψ = θ ∘ ϕ ( c , p , q )
到n n n n = 2 n = 2 n = 2 F 1 ⊗ ( d ) F 2 F_1 \otimes_{(d)} F_2 F 1 ⊗ ( d ) F 2 y 1 ⊗ ( d ) y 2 y_1 \otimes_{(d)} y_2 y 1 ⊗ ( d ) y 2 ⨂ ( d , r , s ) F i \bigotimes_{(d, r, s)} F_i ⨂ ( d , r , s ) F i ⨂ ( d , r , s ) y i \bigotimes_{(d, r, s)} y_i ⨂ ( d , r , s ) y i G G G F 1 ⊗ ( d ) F 2 F_1 \otimes_{(d)} F_2 F 1 ⊗ ( d ) F 2
h : ( x λ ) → ( ⨂ ( c ( 1 ) ) x λ ) ⊗ ( d ) ( ⨂ ( c ( 2 ) ) x λ ) .
h : (x_\lambda) \to \left( \bigotimes_{(c^{(1)})} x_\lambda \right) \otimes_{(d)} \left( \bigotimes_{(c^{(2)})} x_\lambda \right).
h : ( x λ ) → ( c ( 1 ) ) ⨂ x λ ⊗ ( d ) ( c ( 2 ) ) ⨂ x λ . 显然是Z \mathbf{Z} Z ω ∈ Ω \omega \in \Omega ω ∈ Ω ω ∈ Ω 1 \omega \in \Omega_1 ω ∈ Ω 1 ω ∈ Ω 2 \omega \in \Omega_2 ω ∈ Ω 2 ρ ( ω ) ∈ L 1 \rho(\omega) \in L_1 ρ ( ω ) ∈ L 1 σ ( ω ) ∈ L 2 \sigma(\omega) \in L_2 σ ( ω ) ∈ L 2 h ∘ p ˉ ω h \circ \bar{p}_\omega h ∘ p ˉ ω h ∘ q ˉ ω h \circ \bar{q}_\omega h ∘ q ˉ ω ( x λ ) (x_\lambda) ( x λ )
( r ω ( ⨂ ( c ( 1 ) ) x λ ) ) ⊗ ( d ) ( ⨂ ( c ( 2 ) ) x λ ) and ( ⨂ ( c ( 1 ) ) x λ ) ⊗ ( d ) ( s ω ( ⨂ ( c ( 2 ) ) x λ ) )
\left( r_\omega \left( \bigotimes_{(c^{(1)})} x_\lambda \right) \right) \otimes_{(d)} \left( \bigotimes_{(c^{(2)})} x_\lambda \right) \quad \text{and} \quad \left( \bigotimes_{(c^{(1)})} x_\lambda \right) \otimes_{(d)} \left( s_\omega \left( \bigotimes_{(c^{(2)})} x_\lambda \right) \right)
r ω ( c ( 1 ) ) ⨂ x λ ⊗ ( d ) ( c ( 2 ) ) ⨂ x λ and ( c ( 1 ) ) ⨂ x λ ⊗ ( d ) s ω ( c ( 2 ) ) ⨂ x λ 也等于定义或 F 1 ⊗ ( d ) F 2 F_1 \otimes_{(d)} F_2 F 1 ⊗ ( d ) F 2
既然如此，让u u u G G G Z \mathbf{Z} Z H H H Z \mathbf{Z} Z Z \mathbf{Z} Z v : F 1 ⊗ ( d ) F 2 → H v : F_1 \otimes_{(d)} F_2 \to H v : F 1 ⊗ ( d ) F 2 → H u = v ∘ h u = v \circ h u = v ∘ h z 2 = ( x λ ) λ ∈ L 2 z_2 = (x_\lambda)_{\lambda \in L_2} z 2 = ( x λ ) λ ∈ L 2 ∏ λ ∈ L 1 G λ \prod_{\lambda \in L_1} G_\lambda ∏ λ ∈ L 1 G λ H H H
u ( . , z 2 ) : ( x λ ) λ ∈ L 1 ↦ u ( ( x λ ) λ ∈ L 1 , z 2 ) = u ( ( x λ ) λ ∈ L ) .
u(., z_2): (x_\lambda)_{\lambda \in L_1} \mapsto u((x_\lambda)_{\lambda \in L_1}, z_2) = u((x_\lambda)_{\lambda \in L}).
u ( . , z 2 ) : ( x λ ) λ ∈ L 1 ↦ u (( x λ ) λ ∈ L 1 , z 2 ) = u (( x λ ) λ ∈ L ) . 显然它是 Z \mathbf{Z} Z ω ∈ Ω 1 \omega \in \Omega_1 ω ∈ Ω 1 F 1 F_1 F 1 H H H Z \mathbf{Z} Z y 1 ↦ w 1 ( y 1 , x 2 ) y_1 \mapsto w_1(y_1, x_2) y 1 ↦ w 1 ( y 1 , x 2 )
w 1 ( ⨂ ( c ( 1 ) ) x λ , z 2 ) = u ( ( x λ ) λ ∈ L 1 , z 2 )
w_1\left( \bigotimes_{(c^{(1)})} x_\lambda, z_2 \right) = u((x_\lambda)_{\lambda \in L_1}, z_2)
w 1 ( c ( 1 ) ) ⨂ x λ , z 2 = u (( x λ ) λ ∈ L 1 , z 2 ) 我们接下来考虑映射
u 2 : ( x λ ) λ ∈ L 2 ↦ w 1 ( . , ( x λ ) λ ∈ L 2 )
u_2: (x_\lambda)_{\lambda \in L_2} \mapsto w_1(., (x_\lambda)_{\lambda \in L_2})
u 2 : ( x λ ) λ ∈ L 2 ↦ w 1 ( . , ( x λ ) λ ∈ L 2 ) ∏ λ ∈ L 2 G λ \prod_{\lambda \in L_2} G_\lambda ∏ λ ∈ L 2 G λ H o m Z ( F 1 , H ) \mathrm{Hom}_\mathbf{Z}(F_1, H) Hom Z ( F 1 , H ) Z \mathbf{Z} Z u u u ω ∈ Ω 2 \omega \in \Omega_2 ω ∈ Ω 2 ⨂ ( c ( 1 ) ) x λ \bigotimes_{(c^{(1)})} x_\lambda ⨂ ( c ( 1 ) ) x λ Z \mathbf{Z} Z F 1 F_1 F 1 Z \mathbf{Z} Z
w 2 : F 2 → H o m Z ( F 1 , H )
w_2: F_2 \to \mathrm{Hom}_\mathbf{Z}(F_1, H)
w 2 : F 2 → Hom Z ( F 1 , H ) 这样
w 2 ( ⨂ ( c ( 2 ) ) x λ ) = u 2 ( ( x λ ) λ ∈ L 2 )
w_2\left( \bigotimes_{(c^{(2)})} x_\lambda \right) = u_2((x_\lambda)_{\lambda \in L_2})
w 2 ( c ( 2 ) ) ⨂ x λ = u 2 (( x λ ) λ ∈ L 2 ) 或者也
( w 2 ( ⨂ ( c ( 2 ) ) x λ ) ) ( ⨂ ( c ( 1 ) ) x λ ) = u ( ( x λ ) λ ∈ L ) .
\left( w_2\left( \bigotimes_{(c^{(2)})} x_\lambda \right) \right)\left( \bigotimes_{(c^{(1)})} x_\lambda \right) = u((x_\lambda)_{\lambda \in L}).
w 2 ( c ( 2 ) ) ⨂ x λ ( c ( 1 ) ) ⨂ x λ = u (( x λ ) λ ∈ L ) . 我们现在考虑，对于y 1 ∈ F 1 , y 2 ∈ F 2 y_1 \in F_1, y_2 \in F_2 y 1 ∈ F 1 , y 2 ∈ F 2 H H H
w ( y 1 , y 2 ) = ( w 2 ( y 2 ) ) ( y 1 ) .
w(y_1, y_2) = (w_2(y_2))(y_1).
w ( y 1 , y 2 ) = ( w 2 ( y 2 )) ( y 1 ) . 显然，w w w F 1 × F 2 F_1 \times F_2 F 1 × F 2 H H H Z \mathbf{Z} Z ω ∈ Ω ′ \omega \in \Omega' ω ∈ Ω ′ ρ ( ω ) ∈ L 1 \rho(\omega) \in L_1 ρ ( ω ) ∈ L 1 σ ( ω ) ∈ L 2 \sigma(\omega) \in L_2 σ ( ω ) ∈ L 2
w ( r ω ( y 1 ) , y 2 ) = w ( y 1 , s ω ( y 2 ) ) .
w(r_\omega(y_1), y_2) = w(y_1, s_\omega(y_2)).
w ( r ω ( y 1 ) , y 2 ) = w ( y 1 , s ω ( y 2 )) . It suffices to verify this relation when y 1 y_1 y 1 y 2 y_2 y 2 ⨂ ( c ( 1 ) ) x λ \bigotimes_{(c^{(1)})} x_\lambda ⨂ ( c ( 1 ) ) x λ ⨂ ( c ( 2 ) ) x λ \bigotimes_{(c^{(2)})} x_\lambda ⨂ ( c ( 2 ) ) x λ Z \mathbf{Z} Z F 1 F_1 F 1 F 2 F_2 F 2 r ω ( ⨂ ( c ( 1 ) ) x λ ) = ⨂ ( c ( 1 ) ) x λ ′ r_\omega\left( \bigotimes_{(c^{(1)})} x_\lambda \right) = \bigotimes_{(c^{(1)})} x'_\lambda r ω ( ⨂ ( c ( 1 ) ) x λ ) = ⨂ ( c ( 1 ) ) x λ ′ x ρ ( ω ) ′ = p ω ( x ρ ( ω ) ) x'_{\rho(\omega)} = p_\omega(x_{\rho(\omega)}) x ρ ( ω ) ′ = p ω ( x ρ ( ω ) ) x λ ′ = x λ x'_\lambda = x_\lambda x λ ′ = x λ s ω ( ⨂ ( c ( 2 ) ) x λ ) = ⨂ ( c ( 2 ) ) x λ ′ ′ s_\omega\left( \bigotimes_{(c^{(2)})} x_\lambda \right) = \bigotimes_{(c^{(2)})} x''_\lambda s ω ( ⨂ ( c ( 2 ) ) x λ ) = ⨂ ( c ( 2 ) ) x λ ′′ x σ ( ω ) ′ ′ = q ω ( x σ ( ω ) ′ ) x''_{\sigma(\omega)} = q_\omega(x'_{\sigma(\omega)}) x σ ( ω ) ′′ = q ω ( x σ ( ω ) ′ ) x λ ′ ′ = x λ x''_\lambda = x_\lambda x λ ′′ = x λ Z \mathbf{Z} Z v ( y 1 ⊗ y 2 ) = w ( y 1 , y 2 ) v(y_1 \otimes y_2) = w(y_1, y_2) v ( y 1 ⊗ y 2 ) = w ( y 1 , y 2 ) v ∘ h = u v \circ h = u v ∘ h = u
上面定义的一般张量积最重要的特殊情况如下：我们从环的族( A i ) 1 ≤ i ≤ n − 1 (A_i)_{1 \leq i \leq n-1} ( A i ) 1 ≤ i ≤ n − 1 ( E i ) 1 ≤ i ≤ n (E_i)_{1 \leq i \leq n} ( E i ) 1 ≤ i ≤ n E 1 E_1 E 1 A 1 A_1 A 1 E n E_n E n A n − 1 A_{n-1} A n − 1 2 ≤ i ≤ n − 1 2 \leq i \leq n-1 2 ≤ i ≤ n − 1 E i E_i E i ( A i − 1 , A i ) (A_{i-1}, A_i) ( A i − 1 , A i ) [ 1 , n ] [1, n] [ 1 , n ] G i = E i G_i = E_i G i = E i A i A_i A i 1 ≤ i ≤ n − 1 1 \leq i \leq n-1 1 ≤ i ≤ n − 1 ω ∈ A i \omega \in A_i ω ∈ A i 1 ≤ i ≤ n − 1 1 \leq i \leq n-1 1 ≤ i ≤ n − 1 ρ ( ω ) = i \rho(\omega) = i ρ ( ω ) = i σ ( ω ) = i + 1 \sigma(\omega) = i + 1 σ ( ω ) = i + 1 p ω p_\omega p ω Z Z Z E i E_i E i q ω q_\omega q ω x ↦ x ω x \mapsto x\omega x ↦ x ω Z Z Z E i + 1 E_{i+1} E i + 1 y ↦ ω y y \mapsto \omega y y ↦ ω y
E 1 ⊗ A 1 E 2 ⊗ A 2 E 3 ⊗ ⋯ ⊗ A n − 2 E n − 1 ⊗ A n − 1 E n
E_1 \otimes_{A_1} E_2 \otimes_{A_2} E_3 \otimes \cdots \otimes_{A_{n-2}} E_{n-1} \otimes_{A_{n-1}} E_n
E 1 ⊗ A 1 E 2 ⊗ A 2 E 3 ⊗ ⋯ ⊗ A n − 2 E n − 1 ⊗ A n − 1 E n （记号，其中A i A_i A i ⨂ ( c , p , q ) x i \bigotimes_{(c, p, q)} x_i ⨂ ( c , p , q ) x i ( x i ) (x_i) ( x i ) x i ∈ E i x_i \in E_i x i ∈ E i 1 ≤ i ≤ n 1 \leq i \leq n 1 ≤ i ≤ n x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n x_1 \otimes x_2 \otimes \cdots \otimes x_n x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n Z Z Z ⨂ ( c ) v i \bigotimes_{(c)} v_i ⨂ ( c ) v i [ 1 , n ] [1, n] [ 1 , n ] E i E_i E i 2 ≤ i ≤ n − 1 2 \leq i \leq n-1 2 ≤ i ≤ n − 1 n = 3 n = 3 n = 3 Z Z Z E ⊗ A F ⊗ B G E \otimes_A F \otimes_B G E ⊗ A F ⊗ B G
当每个 E i E_i E i 2 ≤ i ≤ n − 1 2 \leq i \leq n-1 2 ≤ i ≤ n − 1 A i − 1 A_{i-1} A i − 1 A i A_i A i i = 1 i = 1 i = 1 i = n i = n i = n E 1 ⊗ A 1 E 2 ⊗ ⋯ ⊗ A n − 1 E n E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n E 1 ⊗ A 1 E 2 ⊗ ⋯ ⊗ A n − 1 E n E i E_i E i 1 ≤ i ≤ n 1 \leq i \leq n 1 ≤ i ≤ n A i A_i A i
特别地，令C为交换环，( E i ) 1 ≤ i ≤ n (E_i)_{1 \leq i \leq n} ( E i ) 1 ≤ i ≤ n E 1 E_1 E 1 E n E_n E n E i E_i E i 2 ≤ i ≤ n − 1 2 \leq i \leq n-1 2 ≤ i ≤ n − 1
E 1 ⊗ C E 2 ⊗ C E 3 ⊗ ⋯ ⊗ C E n − 1 ⊗ C E n
E_1 \otimes_C E_2 \otimes_C E_3 \otimes \cdots \otimes_C E_{n-1} \otimes_C E_n
E 1 ⊗ C E 2 ⊗ C E 3 ⊗ ⋯ ⊗ C E n − 1 ⊗ C E n n n n γ ∈ C \gamma \in C γ ∈ C ( x i ) ∈ ∏ i = 1 n E i (x_i) \in \prod_{i=1}^n E_i ( x i ) ∈ ∏ i = 1 n E i
( γ x 1 ) ⊗ x 2 ⊗ ⋯ ⊗ x n = x 1 ⊗ ( γ x 2 ) ⊗ ⋯ ⊗ x n = ⋯ = x 1 ⊗ x 2 ⊗ ⋯ ⊗ ( γ x n ) .
(\gamma x_1) \otimes x_2 \otimes \cdots \otimes x_n \\
= x_1 \otimes (\gamma x_2) \otimes \cdots \otimes x_n = \cdots = x_1 \otimes x_2 \otimes \cdots \otimes (\gamma x_n).
( γ x 1 ) ⊗ x 2 ⊗ ⋯ ⊗ x n = x 1 ⊗ ( γ x 2 ) ⊗ ⋯ ⊗ x n = ⋯ = x 1 ⊗ x 2 ⊗ ⋯ ⊗ ( γ x n ) . 当我们将张量积 (34) 称为 C 模时，我们总是指的是结构，另外提到的除非，如果不会引起混淆，张量积 (34) 也用 ⨂ 1 ≤ i ≤ n E i \bigotimes_{1 \leq i \leq n} E_i ⨂ 1 ≤ i ≤ n E i ∏ i = 1 n E i \prod_{i=1}^n E_i ∏ i = 1 n E i Z \mathbf{Z} Z
f ( x 1 , … , x i − 1 , γ x i , x i + 1 , … , x n ) = γ f ( x 1 , … , x n )
f(x_1, \ldots, x_{i-1}, \gamma x_i, x_{i+1}, \ldots, x_n) = \gamma f(x_1, \ldots, x_n)
f ( x 1 , … , x i − 1 , γ x i , x i + 1 , … , x n ) = γ f ( x 1 , … , x n ) 对于γ ∈ C \gamma \in \mathbf{C} γ ∈ C ( x i ) ∈ ∏ i E i (x_i) \in \prod_i E_i ( x i ) ∈ ∏ i E i L n ( E 1 , … , E n ; G ) \mathscr{L}_n(E_1, \ldots, E_n; G) L n ( E 1 , … , E n ; G )
L n ( E 1 , … , E n ; G ) → Hom C ( E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n , G )
\mathscr{L}_n(E_1, \ldots, E_n; G) \to \operatorname{Hom}_\mathbf{C}(E_1 \otimes_\mathbf{C} E_2 \otimes \cdots \otimes_\mathbf{C} E_n, G)
L n ( E 1 , … , E n ; G ) → Hom C ( E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n , G ) 它与每个 C-多线性映射 f 和 C-线性映射 g 相关联，使得
f ( x 1 , … , x n ) = g ( x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n ) .
f(x_1, \ldots, x_n) = g(x_1 \otimes x_2 \otimes \cdots \otimes x_n).
f ( x 1 , … , x n ) = g ( x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n ) . E 1 × ⋯ × E n E_1 \times \cdots \times E_n E 1 × ⋯ × E n C \mathbf{C} C n n n
令( F i ) 1 ≤ i ≤ n (F_i)_{1 \leq i \leq n} ( F i ) 1 ≤ i ≤ n u i : E i → F i , u 1 ⊗ u 2 ⊗ ⋯ ⊗ u n u_i : E_i \to F_i, u_1 \otimes u_2 \otimes \cdots \otimes u_n u i : E i → F i , u 1 ⊗ u 2 ⊗ ⋯ ⊗ u n ⨂ 1 ≤ i ≤ n u i \bigotimes_{1 \leq i \leq n} u_i ⨂ 1 ≤ i ≤ n u i
E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n into F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n .
E_1 \otimes_\mathbf{C} E_2 \otimes \cdots \otimes_\mathbf{C} E_n \text{ into } F_1 \otimes_\mathbf{C} F_2 \otimes \cdots \otimes_\mathbf{C} F_n.
E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n into F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n . 此外，( u 1 , … , u n ) ↦ u 1 ⊗ u 2 ⊗ ⋯ ⊗ u n (u_1, \ldots, u_n) \mapsto u_1 \otimes u_2 \otimes \cdots \otimes u_n ( u 1 , … , u n ) ↦ u 1 ⊗ u 2 ⊗ ⋯ ⊗ u n ∏ i Hom C ( E i , F i ) \prod_i \operatorname{Hom}_\mathbf{C}(E_i, F_i) ∏ i Hom C ( E i , F i )
Hom C ( E 1 ⊗ C E ⊗ ⋯ ⊗ C E n , F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n ) .
\operatorname{Hom}_\mathbf{C}(E_1 \otimes_\mathbf{C} E \otimes \cdots \otimes_\mathbf{C} E_n, F_1 \otimes_\mathbf{C} F_2 \otimes \cdots \otimes_\mathbf{C} F_n).
Hom C ( E 1 ⊗ C E ⊗ ⋯ ⊗ C E n , F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n ) . 因此，对应规范地到后者映射，一个C线性映射称为规范
Hom C ( E 1 , F 1 ) ⊗ C Hom C ( E 2 , F 2 ) ⊗ ⋯ ⊗ C Hom C ( E n , F n )
\operatorname{Hom}_\mathbf{C}(E_1, F_1) \otimes_\mathbf{C} \operatorname{Hom}_\mathbf{C}(E_2, F_2) \otimes \cdots \otimes_\mathbf{C} \operatorname{Hom}_\mathbf{C}(E_n, F_n)
Hom C ( E 1 , F 1 ) ⊗ C Hom C ( E 2 , F 2 ) ⊗ ⋯ ⊗ C Hom C ( E n , F n ) → Hom C ( E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n , F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n )
\to \operatorname{Hom}_\mathbf{C}(E_1 \otimes_\mathbf{C} E_2 \otimes \cdots \otimes_\mathbf{C} E_n, F_1 \otimes_\mathbf{C} F_2 \otimes \cdots \otimes_\mathbf{C} F_n)
→ Hom C ( E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n , F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F n ) 将 no. 5 中定义的内容概括为 n = 2 n = 2 n = 2
前面看到的通用结合性性质可以在这里特殊化如下。给定N \mathbf{N} N { 1 , n } \{1, n\} { 1 , n } ( J k ) 1 ≤ k ≤ m (J_k)_{1 \leq k \leq m} ( J k ) 1 ≤ k ≤ m k k k F k F_k F k E i 1 ⊗ C E i 2 ⊗ ⋯ ⊗ C E i r E_{i_1} \otimes_\mathbf{C} E_{i_2} \otimes \cdots \otimes_\mathbf{C} E_{i_r} E i 1 ⊗ C E i 2 ⊗ ⋯ ⊗ C E i r ( i 1 , … , i r ) (i_1, \ldots, i_r) ( i 1 , … , i r ) J k J_k J k
F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F m → E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n
F_1 \otimes_C F_2 \otimes \cdots \otimes_C F_m \to E_1 \otimes_C E_2 \otimes \cdots \otimes_C E_n
F 1 ⊗ C F 2 ⊗ ⋯ ⊗ C F m → E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n 其中，上面的记号、映射张量积
y 1 ⊗ y 2 ⊗ ⋯ ⊗ y m , where y k = x i 1 ⊗ x i 2 ⊗ ⋯ ⊗ x i r ,
y_1 \otimes y_2 \otimes \cdots \otimes y_m, \quad \text{where } y_k = x_{i_1} \otimes x_{i_2} \otimes \cdots \otimes x_{i_r},
y 1 ⊗ y 2 ⊗ ⋯ ⊗ y m , where y k = x i 1 ⊗ x i 2 ⊗ ⋯ ⊗ x i r , 到张量积x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n x_1 \otimes x_2 \otimes \cdots \otimes x_n x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n x i ∈ E i x_i \in E_i x i ∈ E i i i i
特别是，如果π \pi π { 1 , n } \{1, n\} { 1 , n } J k = { π ( k ) } J_k = \{\pi(k)\} J k = { π ( k )} 1 ≤ k ≤ n 1 \leq k \leq n 1 ≤ k ≤ n
E π ( 1 ) ⊗ C E π ( 2 ) ⊗ ⋯ ⊗ C E π ( n ) → E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n
E_{\pi(1)} \otimes_C E_{\pi(2)} \otimes \cdots \otimes_C E_{\pi(n)} \to E_1 \otimes_C E_2 \otimes \cdots \otimes_C E_n
E π ( 1 ) ⊗ C E π ( 2 ) ⊗ ⋯ ⊗ C E π ( n ) → E 1 ⊗ C E 2 ⊗ ⋯ ⊗ C E n 其中映射 x π ( 1 ) ⊗ x π ( 2 ) ⊗ ⋯ ⊗ x π ( n ) x_{\pi(1)} \otimes x_{\pi(2)} \otimes \cdots \otimes x_{\pi(n)} x π ( 1 ) ⊗ x π ( 2 ) ⊗ ⋯ ⊗ x π ( n ) x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n x_1 \otimes x_2 \otimes \cdots \otimes x_n x 1 ⊗ x 2 ⊗ ⋯ ⊗ x n
对于1 ≤ i ≤ n 1 \leq i \leq n 1 ≤ i ≤ n E i E_i E i ( b λ i ( i ) ) λ i ∈ L i (b^{(i)}_{\lambda_i})_{\lambda_i \in L_i} ( b λ i ( i ) ) λ i ∈ L i n n n ( b λ 1 ( 1 ) ⊗ b λ 2 ( 2 ) ⊗ ⋯ ⊗ b λ n ( n ) ) (b^{(1)}_{\lambda_1} \otimes b^{(2)}_{\lambda_2} \otimes \cdots \otimes b^{(n)}_{\lambda_n}) ( b λ 1 ( 1 ) ⊗ b λ 2 ( 2 ) ⊗ ⋯ ⊗ b λ n ( n ) ) ( λ 1 , … , λ n ) (\lambda_1, \ldots, \lambda_n) ( λ 1 , … , λ n ) ∏ 1 ≤ i ≤ n L i \prod_{1 \leq i \leq n} L_i ∏ 1 ≤ i ≤ n L i ⨁ 1 ≤ i ≤ n E i \bigoplus_{1 \leq i \leq n} E_i ⨁ 1 ≤ i ≤ n E i ( b λ 1 ( 1 ) ) (b^{(1)}_{\lambda_1}) ( b λ 1 ( 1 ) )
习题
参见习题为 § 3 。