§ 3. Tensor products

1. Tensor product of two modules

Let $G_1, G_2$ be two $\mathbf{Z}$ -modules; a mapping $u$ of the set $G = G_1 \times G_2$ into a $\mathbf{Z}$ -module is called biadditive (or $\mathbf{Z}$ -bilinear) if $u(x_1, x_2)$ is "additive with respect to $x_1$ and with respect to $x_2$ "; to be precise, this means that, for $x_1, y_1$ in $G_1$ , $x_2, y_2$ in $G_2$ ,

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).

Note that this implies in particular that $u(0, x_2) = u(x_1, 0) = 0$ for all $x_1 \in G_1, x_2 \in G_2$ .

Let A be a ring, E a right A-module and F a left A-module. We are going to consider the universal mapping problem (Set Theory, IV, § 3, no. 1) where $\Sigma$ is the species of $\mathbf{Z}$ -module structure (the morphisms then being $\mathbf{Z}$ -linear mappings, in other words, additive group homomorphisms) and the $\alpha$ -mappings are the mappings $f$ of $E \times F$ into a $\mathbf{Z}$ -module G which are $\mathbf{Z}$ -bilinear and further satisfy, for all $x \in E, y \in F$ and $\lambda \in A$

(1)

f(x \lambda, y) = f(x, \lambda y).

We show that this problem admits a solution. For this we consider the $\mathbf{Z}$ -module $C = \mathbf{Z}^{(E \times F)}$ of formal linear combinations of the elements of $E \times F$ with coefficients in $\mathbf{Z}$ (§ 1, no. 11), a basis of which can be considered to consist of the ordered pairs $(x, y)$ , where $x \in E$ and $y \in F$ . Let D be the sub- $\mathbf{Z}$ -module of C generated by the elements of one of the following types:

(2)

\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}

where $x, x_1, x_2$ are in $E$ , $y, y_1, y_2$ are in $F$ and $\lambda$ is in $A$ .

Definition 1

The tensor product of the right A-module E and the left A-module F*, denoted by $E \otimes_A F$ or $E \otimes_A F$ (or simply $E \otimes F$ if no confusion is to be feared) is the quotient $\mathbf{Z}$ -module $C/D$ (the quotient of the $\mathbf{Z}$ -module C of formal linear combinations of elements of $E \times F$ with coefficients in $\mathbf{Z}$ , by the submodule $D$ generated by the elements of one of the types (2)). *For $x \in E$ and $y \in F$ , the element of $E \otimes_A F$ which is the canonical image of the element $(x, y)$ of $C = \mathbf{Z}^{(E \times F)}$ is denoted by $x \otimes y$ and called the tensor product of $x$ and $y$ .

Cited by 3

Cited by § 3. Tensor products, Remark, Proposition 1

The mapping $(x, y) \mapsto x \otimes y$ of $E \times F$ into $E \otimes_A F$ is called canonical. It is a $\mathbf{Z}$ -bilinear mapping which satisfies conditions (1).

We show that the tensor product $E \otimes_A F$ and the above canonical mapping form a solution of the universal mapping problem posed earlier. To be precise:

Proposition 1

(a) *Let $g$ be a $\mathbf{Z}$ -linear mapping of $E \otimes_A F$ into a $\mathbf{Z}$ -module $G$ . The mapping $(x, y) \mapsto f(x, y) = g(x \otimes y)$ of $E \times F$ into $G$ is $\mathbf{Z}$ -bilinear and satisfies conditions (1).

Cites 3 · Cited by 8

Cites II § 1. Modules, Proposition 17, II § 1, Definition 1

Cited by § 3. Tensor products, Corollary, Proposition 2, Proposition 3, Proposition 8, II § 4. Relations between tensor products and homomorphism modules, Proposition 7, II § 6, Proposition 16, II § 7

(b) Conversely, let $f$ be a $\mathbf{Z}$ -bilinear mapping of $E \times F$ into a $\mathbf{Z}$ -module $G$ satisfying conditions (1). Then there exists one and only one $\mathbf{Z}$ -linear mapping $g$ of $E \otimes_A F$ into $G$ such that $f(x, y) = g(x \otimes y)$ for $x \in E, y \in F$ .

If $\phi$ denotes the canonical mapping of $E \times F$ into $E \otimes_A F$ , then $f = g \circ \phi$ ; whence (a). To show (b), we note that, in the notation of Definition 1, $f$ extends to a $\mathbf{Z}$ -linear mapping $\bar{f}$ of $C$ into $G$ (§ 1, no. 11, Proposition 17). By virtue of relations (1), $\bar{f}$ is zero for all the elements of $C$ of one of the types (2) and hence on $D$ . There therefore exists a $\mathbf{Z}$ -linear mapping $g$ of $C/D = E \otimes_A F$ into $G$ such that $\bar{f} = g \circ \psi$ , where $\psi : C \to C/D$ is the canonical homomorphism (§ 1, no. 8, Remark). The uniqueness of $g$ is immediate since $E \otimes_A F$ is generated, as a $\mathbf{Z}$ -module, by the elements of the form $x \otimes y$ .

Proposition 1 defines a canonical isomorphism of the $\mathbf{Z}$ -module of $\mathbf{Z}$ -bilinear mappings $f$ of $E \times F$ into $G$ , satisfying conditions (1), onto the $\mathbf{Z}$ -module $\mathrm{Hom}_{\mathbf{Z}}(E \otimes_A F, G)$ .

When $A = \mathbf{Z}$ , conditions (1) are automatically satisfied for every $\mathbf{Z}$ -bilinear mapping $f$ and the submodule $D$ of $C$ is already generated by the elements of the first two types in (2).

If now we return to the general case and $E'$ and $F'$ denote the underlying $\mathbf{Z}$ -modules of $E$ and $F$ respectively, the above remark and Definition 1 show immediately that the $\mathbf{Z}$ -module $E \otimes_A F$ can be canonically identified with the quotient of the $\mathbf{Z}$ -module $E' \otimes_{\mathbf{Z}} F'$ by the sub- $\mathbf{Z}$ -module generated by the elements of the form $(x \lambda) \otimes y - x \otimes (\lambda y)$ , where $x$ runs through $E$ , $y$ runs through $F$ and $\lambda$ runs through $A$ .

Corollary 1

Let $H$ be a $\mathbf{Z}$ -module and $h : E \times F \to H$ a $\mathbf{Z}$ -bilinear mapping satisfying conditions (1) and such that $H$ is generated by $h(E \times F)$ . Suppose that for every $\mathbf{Z}$ -module $G$ and every $\mathbf{Z}$ -bilinear mapping $f$ of $E \times F$ into $G$ satisfying (1)* there exists a $\mathbf{Z}$ -linear mapping $g : H \to G$ such that $f = g \circ h$ . Then, if $\phi$ denotes the canonical mapping of $E \times F$ into $E \otimes_A F$ , there exists one and only one isomorphism $\theta$ of $E \otimes_A F$ onto $H$ such that $h = \theta \circ \phi$ .

Cites 1 · Cited by 1

Cites E IV § 3. Universal mappings

Cited by § 3. Tensor products

The hypothesis that $h(E \times F)$ generates $H$ implies the uniqueness of $g$ ; the corollary is then just the general uniqueness property of a solution of a universal mapping problem (Set Theory, IV, § 3, no. 1).

Corollary 2

Let* $E^0$ (resp. $F^0$ ) denote the module $E$ (resp. $F$ ) considered as a left (resp. right) module over the opposite ring $A^0$ ; then there exists one and only one $\mathbf{Z}$ *-module isomorphism* $\sigma : E \otimes_A F \to F^0 \otimes_{A^0} E^0$ such that $\sigma(x \otimes y) = y \otimes x$ for $x \in E$ and $y \in F$ ("commutativity" of tensor products).

Cited by 3

Cited by § 3. Tensor products, Corollary, III § 1. Algebras

By definition of the $A^0$ -module structures on $E^0$ and $F^0$ , the mapping $(x, y) \mapsto y \otimes x$ of $E \times F$ into $F^0 \otimes_{A^0} E^0$ is $\mathbf{Z}$ -bilinear and satisfies conditions (1), whence the existence and uniqueness of the $\mathbf{Z}$ -linear mapping $\sigma$ . Similarly a $\mathbf{Z}$ -linear mapping $\tau : F^0 \otimes_{A^0} E^0 \to E \otimes_A F$ is defined such that $\tau(y \otimes x) = x \otimes y$ and clearly $\sigma$ and $\tau$ are inverse isomorphisms.

Remark

The tensor product of non-zero modules may be zero: for example, taking the two $\mathbf{Z}$ -modules $E = \mathbf{Z}/2\mathbf{Z}$ and $F = \mathbf{Z}/3\mathbf{Z}$ , $2x = 0$ and $3y = 0$ for all $x \in E$ and $y \in F$ ; therefore, in $E \otimes_{\mathbf{Z}} F$ ,

x \otimes y = 3(x \otimes y) - 2(x \otimes y) = x \otimes (3y) - (2x) \otimes y = 0

for all

x

and

y

(cf. no. 6, Corollary 4 to Proposition 6).

Cites 2

Cites Proposition 6, Corollary 4

2. Tensor product of two linear mappings

Let $A$ be a ring, $E, E'$ two right $A$ -modules, $F, F'$ two left $A$ -modules and $u : E \to E'$ and $v : F \to F'$ two $A$ -linear mappings. It is easily verified that the mapping

(x, y) \mapsto u(x) \otimes v(y)

E \times F

into

E' \oplus_A F'

\mathbf{Z}

-bilinear and satisfies conditions (1) of no. 1. By Proposition 1 of no. 1 there thus exists one and only one

\mathbf{Z}

-linear mapping

w : E \otimes_A F \to E' \otimes_A F'

such that

w(x \otimes y) = u(x) \otimes v(y)

for

x \in E, y \in F

. This mapping is denoted by

u \otimes v

(when no confusion can arise) and is called the tensor product of the linear mappings

u

and

v

It follows immediately from (3) that $(u, v) \mapsto u \otimes v$ is a $\mathbf{Z}\text{-bilinear}$ mapping called canonical

\operatorname{Hom}_A(E, E') \times \operatorname{Hom}_A(F, F') \to \operatorname{Hom}_{\mathbf{Z}}(E \otimes_A F, E' \otimes_A F').

There corresponds to it by Proposition 1 of no. 1 a $\mathbf{Z}$ -linear mapping called canonical

(4)

\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')

which associates with every element

u \otimes v

of the tensor product the linear mapping

u \otimes v : E \otimes_A F \to E' \otimes_A F'

. Note that the canonical mapping (4) is not necessarily injective nor surjective. The notation

u \otimes v

can therefore lead to confusion and it will be necessary for the context to indicate whether it denotes an element of the tensor product or a linear mapping.

Further, let $E''$ be a right $A$ -module, $F''$ a left $A$ -module and $u' : E' \to E''$ , $v' : F' \to F''$ $A$ -linear mappings; it follows from (3) that (5)

(u' \circ u) \otimes (v' \circ v) = (u' \otimes v') \circ (u \otimes v).

3. Change of ring

Proposition 2

Let $A, B$ be two rings, $\rho : B \to A$ a ring homomorphism and $E$ (resp. $F$ ) a right (resp. left) $A$ -module. Then there exists one and only one $\mathbf{Z}$ -linear mapping* (6)

\phi : \rho_*(E) \otimes_B \rho_*(F) \to E \otimes_A F

*such that, for all

x \in E

and

y \in F

, the image under

\phi

of the element

x \otimes y

\rho_*(E) \otimes_B \rho_*(F)

is the element

x \otimes y

E \otimes_A F

; this

\mathbf{Z}

-linear mapping is surjective.

Cites 1 · Cited by 1

Cites Proposition 1

Cited by Corollary

We consider the mapping $(x, y) \mapsto x \otimes y$ of $\rho_*(E) \times \rho_*(F)$ into $E \otimes_A F$ ; it is $\mathbf{Z}$ -bilinear and, for all $\beta \in B$ , by definition $(x \rho(\beta)) \otimes y = x \otimes (\rho(\beta)y)$ , hence conditions (1) of no. 1 hold, whence the existence and uniqueness of $\phi$ (no. 1, Proposition 1). The latter assertion follows from the fact that the elements $x \otimes y$ generate the $\mathbf{Z}$ -module $E \otimes_A F$ .

The mapping (6) is called canonical.

Corollary

Let $\mathfrak{J}$ be a two-sided ideal of $A$ such that $\mathfrak{J}$ is contained in the annihilator of $E$ and in the annihilator of $F$ , so that $E$ (resp. $F$ ) has a canonical left (resp. right) $(A/\mathfrak{J})$ -module structure* ( $§ 1$ , no. 12). Then the canonical homomorphism (6)

\phi : E \otimes_A F \to E \otimes_{A/\mathfrak{J}} F

*corresponding to the canonical homomorphism

\rho : A \to A/\mathfrak{J}

is the identity.

Cites 1 · Cited by 1

Cites Proposition 2

Cited by Corollary 3

For all $\bar{\alpha} \in A/\mathfrak{J}$ , all $x \in E$ and all $y \in F$ , $x \bar{\alpha} = x \alpha$ (resp. $\bar{\alpha} y = \alpha y$ ) for all $\alpha$ such that $\rho(\alpha) = \bar{\alpha}$ . If $C = \mathbf{Z}^{(E \times F)}$ , the submodule of $C$ generated by the elements $(x \alpha, y) - (x, \alpha y)$ is then equal to the submodule generated by the elements $(x \bar{\alpha}, y) - (x, \bar{\alpha} y)$ .

With the hypotheses and notation of Proposition 2, let $E'$ be a right $B$ -module, $F'$ a left $B$ -module and consider two semi-linear mappings $u : E' \to E$ , v:F' \to F relative to the homomorphism $\rho : B \to A$ ; $u$ (resp. $v$ ) can be considered as a $B$ -linear mapping $E' \to \rho_*(E)$ (resp. $F' \to \rho_*(F')$ ), whence a $\mathbf{Z}$ -linear mapping $w : E' \otimes_B F' \to \rho_*(E) \otimes_B \rho_*(F)$ such that $w(x' \otimes y') = u(x') \otimes v(y')$ for $x' \in E',\ y' \in F'$ ; composing the canonical mapping (6) with this mapping, a $\mathbf{Z}$ -linear mapping $w' : E' \otimes_B F' \to E \otimes_A F$ is hence obtained such that $w'(x' \otimes y') = u(x') \otimes v(y')$ for $x' \in E',\ y' \in F'$ ; this is the mapping which will normally be denoted by $u \otimes v$ if no confusion can arise. Clearly $(u, v) \mapsto u \otimes v$ is a $\mathbf{Z}$ -bilinear mapping

\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).

Moreover, if $C$ is a third ring, $\sigma : C \to B$ a homomorphism, $E''$ a right $C$ -module, $F''$ a left $C$ -module, $u' : E'' \to E'$ and $v' : F'' \to F'$ semi-linear mappings relative to $\sigma$ , then

(u \circ u') \otimes (v \circ v') = (u \otimes v) \circ (u' \otimes v').

4. Operators on a tensor product; tensor products as multimodules

With the hypotheses and notation of no. 1, for every endomorphism $u$ (resp. $v$ ) of the $A$ -module $E$ (resp. $F$ ), $u \otimes 1_F$ (resp. $1_E \otimes v$ ) is an endomorphism of the $\mathbf{Z}$ -module $E \otimes_A F$ ; it follows immediately from (5) (no. 2) that the mapping $u \mapsto u \otimes 1_F$ (resp. $v \mapsto 1_E \otimes v$ ) is a ring homomorphism

\operatorname{End}_A(E) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F)

(resp. $\operatorname{End}_A(F) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F)$ ); moreover,

(u \otimes 1_F) \circ (1_E \otimes v) = (1_E \otimes v) \circ (u \otimes 1_F) = u \otimes v

and therefore (§ 1, no. 14) $E \otimes_A F$ has a canonical left bimodule structure with respect to the rings $\operatorname{End}_A(E)$ and $\operatorname{End}_A(F)$ .

This being so, suppose given on $E$ a $((B'_i); A, (C'_j))$ -multimodule structure and on $F$ a $(A, (B''_h); (C''_{k_c}))$ -multimodule structure (§ 1, no. 14); it amounts to the same to say that ring homomorphisms $B'_i \to \operatorname{End}_A(E)$ , $C'_j{}^0 \to \operatorname{End}_A(E)$ are given with pairwise permutable images, and ring homomorphisms $B''_h \to \operatorname{End}_A(F)$ , $C''_{k_c}{}^0 \to \operatorname{End}_A(F)$ with pairwise permutable images. If these homomorphisms are composed respectively with the canonical homomorphisms $\operatorname{End}_A(E) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F)$ and $\operatorname{End}_A(F) \to \operatorname{End}_\mathbf{Z}(E \otimes_A F)$ defined above, it is seen (taking account of (7)) that ring homomorphisms

\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*}

are defined with pairwise permutable images; in other words, there has been defined on $E \otimes_A F$ a $((B'_i), (B''_h); (C'_j), (C''_{k_c}))$ -multimodule structure; it is this multimodule which is also called the tensor product (relative to $A$ ) of the

((B'_i); A, (C'_j))-multimodule E and the (A, (B''_h); (C''_k))-multimodule F. This multimodule is the solution of a universal mapping problem analogous to that considered in no. 1; to be precise:

Proposition 3

Let G be a ((B'_i), (B''_h); (C'_j), (C''_k))-multimodule.

Cites 1 · Cited by 2

Cites Proposition 1

Cited by § 3. Tensor products, Scholium

(a) Let g be a linear mapping of the multimodule E $\otimes_A$ F into G. The mapping $f : (x, y) \mapsto g(x \otimes y)$ of E $\times$ F into G is $\mathbf{Z}$ -bilinear and satisfies relations (1) of no. 1 and the conditions

\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}

for all $x \in E, y \in F, \mu'_i \in B'_i, v'_j \in C'_j, \mu''_h \in B''_h, v''_k \in C''_k, i, j, h, k$ arbitrary.

(b) Conversely, let f be a $\mathbf{Z}$ -bilinear mapping of E $\times$ F into G satisfying conditions (1) (no. 1) and (8). Then there exists one and only one linear mapping g of the multimodule E $\otimes_A$ F into the multimodule G such that $f(x, y) = g(x \otimes y)$ for $x \in E, y \in F$ .

Assertion (a) follows immediately from the definition of the multimodule structure on E $\otimes_A$ F, since for example $(x \otimes y)v'_j = (xv'_j) \otimes y$ . To prove (b), we note first that Proposition 1 of no. 1 gives the existence and uniqueness of a $\mathbf{Z}$ -linear mapping g such that $g(x \otimes y) = f(x, y)$ for $x \in E, y \in F$ ; all that is needed is to verify that g is linear for the multimodule structures. As the elements $x \otimes y$ generate the $\mathbf{Z}$ -module E $\otimes_A$ F, it suffices to verify the relations $g(\mu'(x \otimes y)) = \mu'_i g(x \otimes y)$ and their analogues; but this follows immediately from the formula $g(x \otimes y) = f(x, y)$ and relations (8).

Scholium

An element of E $\otimes_A$ F may in general be written in several ways in the form $\sum_i (x_i \otimes y_i)$ , where $x_i \in E$ and $y_i \in F$ ; but to define a linear mapping g of the multimodule E $\otimes_A$ F into a multimodule G, there is no need to verify that, if $\sum_i (x_i \otimes y_i) = \sum_j (x'_j \otimes y'_j)$ , then $\sum_i g(x_i \otimes y_i) = \sum_j g(x'_j \otimes y'_j)$ ; it suffices to be given $g(x \otimes y)$ for $x \in E$ and $y \in F$ and to verify that $(x, y) \mapsto g(x \otimes y)$ is $\mathbf{Z}$ -bilinear and satisfies conditions (1) (no. 1) and (8).

Let E' be a ((B'_i), A, (C'_j))-multimodule, F' an (A, (B''_h); (C''_k))-multimodule and $u : E \to E', v : F \to F'$ linear mappings of multimodules; it follows immediately from the definitions (no. 2) that $u \otimes v$ is a linear mapping of the multimodule E $\otimes_A$ F into the multimodule E' $\otimes_A$ F'.

With E always denoting a right A-module, let $_sA_d$ denote the ring A considered as an (A, A)-bimodule (§ 1, no. 14, Example 1); by the above, the tensor product E $\otimes_A (_sA_d)$ has a canonical right A-module structure such that $(x \otimes \lambda)\mu = x \otimes (\lambda \mu)$ for $x \in E, \lambda \in A, \mu \in A$ . The mapping $(x, \lambda) \mapsto x \lambda$ of E $\times (_sA_d)$ into E is $\mathbf{Z}$ -bilinear and satisfies conditions (1) (no. 1) and (8) (where, in the latter, the $B'_i, C'_j$ and $B''_h$ are absent and the family $(C''_k)$ reduces to $A$ ); hence (Proposition 3), there exists an $A$ -linear mapping $g$ (called canonical) of $E \otimes_A (sA_d)$ into $E$ such that $g(x \otimes \lambda) = x\lambda$ for $x \in E, \lambda \in A$ .

Cites 2

Cites Example, II § 1, Proposition 3

Proposition 4

If $E$ is a right $A$ -module, the mapping $h : x \mapsto x \otimes 1$ of $E$ into $E \otimes_A (sA_d)$ is a right $A$ -module isomorphism, whose inverse isomorphism $g$ is such that $g(x \otimes \lambda) = x\lambda$ for $x \in E, \lambda \in A$ .

Cited by 16

Cited by AC I § 2. Flat modules, Proposition 7, II § 11, Remark, Corollary 1, Corollary 3, Corollary 6, Lemma 3, II § 4, Proposition 4, II § 4, Corollary, II § 5, Proposition 2, II § 5, Proposition 6, II § 5, Proposition 26, II § 7, III § 1. Algebras, Proposition 3, III § 11, Proposition 13, III § 7, Definition 1, VIII § 6

If $g$ is the canonical mapping, $g \circ h$ is the identity mapping $1_E$ and $h \circ g$ coincides with the identity mapping of $E \otimes_A (sA_d)$ onto itself for elements of the form $x \otimes y$ , which generate the latter $\mathbf{Z}$ -module; hence the conclusion.

We shall normally write $E \otimes_A A$ instead of $E \otimes_A (sA_d)$ and shall often identify $E \otimes_A A$ with $E$ by means of the above canonical isomorphisms. Observe that, if $E$ also has a (left or right) $B$ -module structure which is compatible with its right $A$ -module structure, $g$ and $h$ are also isomorphisms for the $B$ -module structures on $E$ and $E \otimes_A A$ (and hence multinode module isomorphisms).

Now let $F$ be a left $A$ -module; $(sA_d) \otimes_A F$ (also denoted by $A \otimes_A F$ ) then has a canonical left $A$ -module structure and as in Proposition 4 a canonical isomorphism is defined of $A \otimes_A F$ onto $F$ mapping $\lambda \otimes x$ to $\lambda x$ , and its inverse isomorphism is $x \mapsto 1 \otimes x$ .

In particular there exists a canonical isomorphism of the $(A, A)$ -bimodule $(sA_d) \otimes_A (sA_d)$ onto $sA_d$ which maps $\lambda \otimes \mu$ to $\lambda \mu$ .

5. Tensor product of two modules over a commutative ring

Let $C$ be a commutative ring; for every $C$ -module $E$ , the module structure on $E$ is compatible with itself (§ 1, no. 14). If $E$ and $F$ are two $C$ -modules, the considerations of no. 4 then allow us to define two $C$ -module structures on the tensor product $E \otimes_C F$ , respectively such that $\gamma(x \otimes y) = (\gamma x) \otimes y$ and such that $\gamma(x \otimes y) = x \otimes (\gamma y)$ ; but as, by Definition 1 of no. 1, in this case $(\gamma x) \otimes y = x \otimes (\gamma y)$ , these two structures are the same. Henceforth when we speak of $E \otimes_C F$ as a $C$ -*module*, we shall mean with the structure thus defined, unless otherwise stated. The canonical isomorphism.

\sigma : F \otimes_C E \to E \otimes_C F

(no. 1, Corollary 2 to Proposition 1) is then a $C$ -module isomorphism.

It follows from this definition that, if $(a_\lambda)_{\lambda \in L}$ (resp. $(b_\mu)_{\mu \in M}$ ) is a generating system of the $C$ -module $E$ (resp. $F$ ), $(a_\lambda \otimes b_\mu)$ is a generating system of the $C$ -module $E \otimes_C F$ ; in particular, if $E$ and $F$ are finitely generated $C$ -modules, so is $E \otimes_C F$ .

For every $C$ -module $G$ , the $\mathbf{Z}$ -bilinear mappings $f$ of $E \times F$ into $G$ for which

f(\gamma x, y) = f(x, \gamma y) = \gamma f(x, y) \quad \text{for } x \in E, y \in F, \gamma \in C

are then called C-bilinear and form a C-module denoted by $\mathscr{L}_2(E, F; G)$ ; Proposition 3 (no. 4) defines a canonical C-module isomorphism (cf. § 1, no. 14, Remark 1).

\mathscr{L}_2(E, F; G) \to \mathrm{Hom}_C(E \otimes_C F, G).

Let E', F' be two C-modules and $u : E \to E'$ , $v : F \to F'$ two C-linear mappings; then (no. 4) $u \otimes v$ is a C-linear mapping of $E \otimes_C F$ into $E' \otimes_C F'$ . Further, it is immediate that $(u, v) \mapsto u \otimes v$ is a C-bilinear mapping of $\mathrm{Hom}_C(E, E') \times \mathrm{Hom}_C(F, F')$ into $\mathrm{Hom}_C(E \otimes_C F, E' \otimes_C F')$ ; hence there canonically corresponds to it a C-linear mapping, called canonical:

\mathrm{Hom}_C(E, E') \otimes_C \mathrm{Hom}_C(F, F') \to \mathrm{Hom}_C(E \otimes_C F, E' \otimes_C F')

which associates with every element $u \otimes v$ of the tensor product

\mathrm{Hom}_C(E, E') \otimes_C \mathrm{Hom}_C(F, F')

the linear mapping $u \otimes v$ . Note that the canonical mapping (11) is not necessarily injective nor surjective (§ 4, Exercise 2).

Remark

(1) Let A, B be two commutative rings, $\rho : B \to A$ a ring homomorphism and E and F two A-modules; then the canonical mapping (6) of no. 3 is a B-linear mapping

\rho_*(E) \otimes \rho_*(F) \to \rho_*(E \otimes_A F).

(2) What has been said in this no. can be generalized to the following case: let E be a right A-module, F a left A-module, C a commutative ring and $\rho : C \to A$ a homomorphism of C into A such that $\rho(C)$ is contained in the centre of A (cf. III, § 1, no. 3). We may then consider the C-modules $\rho_*(E)$ and $\rho_*(F)$ and the hypothesis on $\rho$ implies that these C-module structures are compatible respectively with the A-module structures on E and F (§ 1, no. 14). The tensor product $E \otimes_A F$ is thus (by virtue of no. 4) given two C-module structures such that $\gamma(x \otimes y) = (x \rho(\gamma)) \otimes y$ and

\gamma(x \otimes y) = x \otimes (\rho(\gamma)y)

respectively for $\gamma \in C$ , $x \in E$ , $y \in F$ and Definition 1 (no. 1) shows also that these two structures are identical. If E' (resp. F') is a right (resp. left) A-module and $u : E \to E'$ , $v : F \to F'$ are two A-linear mappings, then $u \otimes v : E \otimes_A F \to E' \otimes_A F'$ is C-linear for the C-module structures just defined; the mapping $(u, v) \mapsto u \otimes v$ :

\mathrm{Hom}_A(E, E') \times \mathrm{Hom}_A(F, F') \to \mathrm{Hom}_C(E \otimes_A F, E' \otimes_A F')

is C-bilinear (for the C-module structures on $\mathrm{Hom}_A(E, E')$ and $\mathrm{Hom}_A(F, F')$ ) defined in § 1, no. 14, Remark 1), whence we also derive a C-linear mapping, called *canonical

(13) $\operatorname{Hom}_A(E, E') \otimes_C \operatorname{Hom}_A(F, F') \to \operatorname{Hom}_C(E \otimes_A F, E' \otimes_A F')$ .

(3) Let A be an integral domain and K its field of fractions. If E and F are two vector K-spaces, the canonical mapping

(E_{[A]}) \otimes_A (F_{[A]}) \to E \otimes_K F

(no. 3 and § 1, no. 13) is bijective. It suffices (no. 4) to prove that if $f$ is an A-bilinear mapping of $E \times F$ into a vector K-space G, $f$ is also K-bilinear. Now, for all $\alpha \neq 0$ in A.

\alpha f(\alpha^{-1}x, y) = f(x, y) = \alpha f(x, \alpha^{-1}y)

whence

f(\alpha^{-1}x, y) = f(x, \alpha^{-1}y) = \alpha^{-1}f(x, y)

since G is a vector K-space.

Cites 3

Cites II § 1. Modules, III § 1. Algebras, Definition 1

6. Properties of $E \otimes_A F$ relative to exact sequences

Proposition 5

Let* E, E', E'' be right A-*modules*, F a left A-*module* and

(14)

E' \xrightarrow{u} E \xrightarrow{v} E'' \to 0

an exact sequence of linear mappings.* Writing $\bar{u} = u \otimes 1_F, \bar{v} = v \otimes 1_F,$ *the sequence

Cited by 8

Cited by Proposition 7, II § 11, Corollary, Proposition 6, Exercise 6, § 4, Proposition 14, II § 7, VIII § 11. Grothendieck Groups, Proposition 9, VIII § 11, Proposition 11, VIII § 6

(15)

E' \otimes_A F \xrightarrow{\bar{u}} E \otimes_A F \xrightarrow{\bar{v}} E'' \otimes_A F \to 0

of $\mathbf{Z}$ -*homomorphisms* is exact.

By virtue of no. 2, formula (5), $\bar{v} \circ \bar{u} = (v \circ u) \otimes 1_F = 0$ ; the image $H = \bar{u}(E' \otimes F)$ is contained in the kernel $L = \operatorname{Ker}(\bar{v})$ ; by passing to the quotient, we therefore derive from $\bar{v}$ a $\mathbf{Z}$ -linear mapping $f$ of the cokernel $M = (E \otimes F)/H$ of $\bar{u}$ into $E'' \otimes F$ ; it must be proved that $f$ is bijective and it will hence suffice to define a $\mathbf{Z}$ -linear mapping $g : E'' \otimes F \to M$ such that $g \circ f$ and $f \circ g$ are the identity mappings.

Let $x'' \in E''$ , $y \in F$ ; by hypothesis there exists $x \in E$ such that $v(x) = x''$ . We show that, if $x_1, x_2$ are two elements of E such that $v(x_1) = v(x_2) = x''$ and $\phi : E \otimes F \to M$ is the canonical mapping, then $\phi(x_1 \otimes y) = \phi(x_2 \otimes y)$ . It suffices to prove that if $v(x) = 0$ then $\phi(x \otimes y) = 0$ , which follows from the fact that $x = u(x')$ with $x' \in E'$ , whence $x \otimes y = u(x') \otimes y = \bar{u}(x' \otimes y) \in H$ . If $(x'', y)$ is mapped to the unique value of $\phi(x \otimes y)$ for all $x \in E$ such that $v(x) = x''$ , a mapping is defined of $E'' \times F$ into M; this mapping is $\mathbf{Z}$ -bilinear and satisfies conditions (1) (no. 1), since $v(x \lambda) = x'' \lambda$ and $(x \lambda) \otimes y = x \otimes (\lambda y)$ for $x \in E$ ; hence there is a $\mathbf{Z}$ -linear mapping $g$ of $E'' \otimes F$ into M such that $g(x'' \otimes y) = \phi(x \otimes y)$ for $y \in F, x \in E$ and $x'' = v(x)$ . This definition further proves that $f \circ g$ coincides with the identity mapping for the elements of

$E'' \otimes F$ of the form $x'' \otimes y$ and hence $f \circ g$ is the identity mapping of $E'' \otimes F$ ; on the other hand, for $x \in E$ and $y \in F$ , $f(\phi(x \otimes y)) = v(x) \otimes y$ by definition, hence $g(f(\phi(x \otimes y))) = \phi(x \otimes y)$ and, as the elements of the form $\phi(x \otimes y)$ generate M, $g \circ f$ is the identity mapping of M.

Corollary

Let F, F', F'' be left A-modules, E a right A-module and*

\begin{array}{ccccc} F' & \xrightarrow{s} & F & \xrightarrow{t} & F'' \longrightarrow 0 \end{array}

(16) an exact sequence of linear mappings. Writing \bar{s} = lE \otimes s, \bar{t} = l_E \otimes t, the sequence of \mathbf{Z}-homomorphisms*

\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}

(17) *is exact.

Cites 3 · Cited by 2

Cites Proposition 1, Corollary 2, Proposition 5

Cited by Lemma 1, AC I § 2, Proposition 6

When E (resp. F) is considered as a left (resp. right) A^0-module, F \otimes_{A^0} E is identified with E \otimes_A F and there are analogous identifications for F' \otimes_{A^0} E and F'' \otimes_{A^0} E (no. 1, Corollary 2 to Proposition 1); the corollary then follows immediately from Proposition 5.

Remark

Note that in general, if E' is a submodule of a right A-module E and j : E' \to E the canonical injection, the canonical mapping

j \otimes 1_F : E' \otimes F \to E \otimes F

is not necessarily injective. In other words, for an exact sequence

\begin{array}{ccccccc} 0 & \longrightarrow & E' & \xrightarrow{u} & E & \xrightarrow{v} & E'' \longrightarrow 0 \end{array}

(18) it cannot in general be concluded that the sequence

\begin{array}{ccccccc} 0 & \longrightarrow & E' \otimes F & \xrightarrow{u} & E \otimes F & \xrightarrow{v} & E'' \otimes F \longrightarrow 0 \end{array}

(19) *is exact.

Take for example A = \mathbf{Z}, E = \mathbf{Z}, E' = 2\mathbf{Z}, F = \mathbf{Z}/2\mathbf{Z}. As E' is isomorphic to E, E' \otimes F is isomorphic to E \otimes F which is itself isomorphic to F (no. 4, Proposition 4). But for all x' = 2x \in E' (where x \in E) and all y \in F, j(x') \otimes y = (2x) \otimes y = x \otimes (2y) = 0, since 2y = 0, and the canonical image of E' \otimes F in E \otimes F reduces to 0.

In other words, care must be taken to distinguish, for a submodule E' of E and an element x \in E', between the element x \otimes y "calculated in E' \otimes F'" and the element x \otimes y "calculated in E \otimes F" (in other words, the element j(x) \otimes y).

We shall study later, under the name of flat modules, the modules F such that the sequence (19) is exact for every exact sequence (18) (Commutative Algebra, I, § 2).

Cites 1

Cites Proposition 4

Proposition 6

Given two exact sequences (14) and (16), the homomorphism v \otimes t : E \otimes_A F \to E'' \otimes_A F'' is surjective and its kernel is equal to*

\operatorname{Im}(u \otimes 1_F) + \operatorname{Im}(l_E \otimes s)

Cites 2 · Cited by 4

Cites Proposition 5, Corollary

Cited by Corollary 3, II § 11, Remark, Exercise 1, § 1, Proposition 3, III § 5

Now $v \otimes t = (v \otimes 1_{F''}) \circ (1_E \otimes t)$ (no. 2, formula (5)) and $v \otimes t$ is therefore surjective, being the composition of two surjective homomorphisms by virtue of Proposition 5 and its Corollary. On the other hand, for $z \in E \otimes F$ to be in the kernel of $v \otimes t$ , it is necessary and sufficient that $(1_E \otimes t)(z)$ belong to the kernel of $v \otimes 1_{F''}$ , that is, by virtue of (15) to the image of

u \otimes 1_{F''}: E' \otimes F'' \to E \otimes F''.

But as the homomorphism $t: F \to F''$ is surjective, so is

1_{E'} \otimes t: E' \otimes F \to E' \otimes F''

by the Corollary to Proposition 5, hence the condition on $z$ reduces to the existence of an $a \in E' \otimes F$ such that

(l_E \otimes t)(z) = (u \otimes t)(a).

Let $b = z - (u \otimes 1_F)(a)$ ; then $(l_E \otimes t)(b) = 0$ , and by virtue of (17), $b$ belongs to the image of $l_E \otimes s$ , which proves the proposition.

In other words:

Corollary 1

Let $E'$ be a submodule of a right $A$ -module $E$ , $F'$ a submodule of a left $A$ -module $F$ and $\operatorname{Im}(E' \otimes_A F)$ and $\operatorname{Im}(E \otimes_A F')$ the sub- $\mathbf{Z}$ -modules of $E \otimes_A F$ , the respective images of the canonical mappings $E' \otimes_A F \to E \otimes_A F$ ,

E \otimes_A F' \to E \otimes_A F.

Then there is a canonical $\mathbf{Z}$ -module isomorphism

Cited by 2

Cited by Proposition 1, III § 4, Proposition 8, IV § 6

(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'))

*such that, for $\xi \in E/E'$ , $\eta \in F$ , $\pi(\xi \otimes \eta)$ is the class of all elements $x \otimes y \in E \otimes_A F$ such that $x \in \xi$ and $y \in \eta$ .

Note that when $E$ is a $((B'_i); A, (C'_j))$ -multimodule, $F$ a $(A, (B''_h); (C''_k))$ -multimodule and $E'$ and $F'$ submultimodules of $E$ and $F$ respectively, the isomorphism (20) is an isomorphism for the $((B'_i), (B''_h); (C'_j), (C''_k))$ -multimodule structures of the two sides (no. 3).

Corollary 2

Let $a$ be a right ideal of $A$ , $F$ a left $A$ -module and $aF$ the sub- $\mathbf{Z}$ -module of $F$ generated by the elements of the form $\lambda x$ , where $\lambda \in a$ and $x \in F$ . Then there is a canonical $\mathbf{Z}$ -module isomorphism

Cited by 1

Cited by Proposition 7, II § 11

(21)\quad \pi : (A/a) \otimes_A F \to F/aF

*such that, for all $\bar{\lambda} \in A/a$ and all $x \in F$ , $\pi(\bar{\lambda} \otimes x)$ is the class mod. $aF$ of $\lambda x$ , where $\lambda \in \bar{\lambda}$ .

In particular, for $A = \mathbf{Z}$ , it is seen that for every integer $n$ and every $\mathbf{Z}$ -module $F$ , $(\mathbf{Z}/n\mathbf{Z}) \otimes_{\mathbf{Z}} F$ is canonically identified with the quotient $\mathbf{Z}$ -module $F/nF$ .

Corollary 3

Let $A$ be a commutative ring, $a$ an ideal of $A$ and $E$ and $F$ two $A$ -modules such that $a$ is contained in the annihilator of $F$ . Then the $(A/a)$ -modules $E \otimes_A F$ and $(E/aE) \otimes_{A/a} F$ are canonically isomorphic.

Cites 1 · Cited by 1

Cites Corollary

Cited by Proposition 1, AC III § 5

$F$ and $E \otimes_A F$ are annihilated by $a$ and hence have canonical $(A/a)$ -module structures ( $§ 1$ , no. 12) and if we write $E' = aE$ , then $\operatorname{Im}(E' \otimes_A F) = 0$ ; then there is a canonical isomorphism (20) of $E \otimes_A F$ onto $(E/aE) \otimes_A F$ and the latter is itself identical with $(E/aE) \otimes_{A/a} F$ (no. 3, Corollary to Proposition 2).

Corollary 4

Let $a, b$ be two ideals in a commutative ring $C$ ; the $C$ -module $(C/a) \otimes_C (C/b)$ is then canonically isomorphic to $C(a + b)$ .

Cited by 1

Cited by Remark

7. Tensor products of products and direct sums

Let $(E_\lambda)_{\lambda \in L}$ be a family of right $A$ -modules, $(F_\mu)_{\mu \in M}$ a family of left $A$ -modules and consider the product modules $C = \prod_{\lambda \in L} E_\lambda, D = \prod_{\mu \in M} F_\mu$ . The mapping $((x_\lambda), (y_\mu)) \mapsto (x_\lambda \otimes y_\mu)$ of $C \times D$ into the product $\mathbf{Z}$ -module

\prod_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu)

\mathbf{Z}

-bilinear and obviously satisfies conditions (1) (no. 1). Thus there exists (no. 1, Proposition 1) a

\mathbf{Z}

-linear mapping, called canonical

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)

such that

f((x_\lambda) \otimes (y_\mu)) = (x_\lambda \otimes y_\mu)

When $C = R^L, D = S^M, R$ (resp. $S$ ) being a right (resp. left) $A$ -module, the canonical mapping (22) associates with every tensor product $u \otimes v$ , where $u$ is a mapping of $L$ into $R$ and $v$ a mapping of $M$ into $S$ , the mapping $(\lambda, \mu) \mapsto u(\lambda) \otimes v(\mu)$ of $L \times M$ into $R \otimes_A S$ ; even in this case the canonical mapping (22) is in general neither injective nor surjective (Exercise 3; cf. Corollary 3 to Proposition 7).

When the $E_\lambda$ are $((B'_i); A, (C'_j))$ -multimodules and the $F_\mu$ $(A, (B''_h); (C''_{k_c}))$ -multimodules, the homomorphism (22) is also a homomorphism for the $((B'_i), (B''_h); (C'_j), (C''_{k_c}))$ -multimodule structures of the two sides.

Consider now the submodule $E = \bigoplus_{\lambda \in L} E_\lambda$ (resp. $\bigoplus_{\mu \in M} F_\mu$ ) of $C$ (resp. $D$ ); the canonical injections $E \to C, F \to D$ define canonically a $\mathbf{Z}$ -linear mapping $E \otimes_A F \to C \otimes_A D$ which, composed with the mapping (22), gives a $\mathbf{Z}$ -linear mapping $g$ of $E \otimes_A F$ into $\prod_{\lambda, \mu} (E_\lambda \otimes_A F_\mu)$ such that

g((x_\lambda) \otimes (y_\mu)) = (x_\lambda \otimes y_\mu);

moreover, as the families $(x_\lambda)$ and $(y_\mu)$ have finite support, so does $(x_\lambda \otimes y_\mu)$ and hence finally $g$ is a canonical homomorphism

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),

which is a multimodule homomorphism under the same conditions as (22).

Proposition 7

The canonical mapping (23) is bijective.

Cites 1 · Cited by 15

Cites Proposition 6, II § 1

Cited by Proposition 2, AC I § 2, II § 11. Graded modules and rings, § 3. Tensor products, Remark 2, Corollary 1, Corollary 3, Corollary 5, Corollary 6, Lemma 2, II § 4, Lemma 5, II § 4, Corollary, II § 5, Proposition 2, III § 4, III § 5. Tensor algebra. Tensors, Lemma 2, VIII § 15, Proposition 5, VIII § 7

To prove this it suffices to define a $\mathbf{Z}$ -linear mapping $h$ of the direct sum $G = \bigoplus_{(\lambda, \mu) \in L \times M} (E_\lambda \otimes_A F_\mu)$ into $E \otimes_A F$ such that $g \circ h$ and $h \circ g$ are the identity mappings. But, to define a $\mathbf{Z}$ -linear mapping of $G$ into $E \otimes_A F$ , it suffices (§ 1, no. 6, Proposition 6) to define a $\mathbf{Z}$ -linear mapping

h_{\lambda \mu} : E_\lambda \otimes_A F_\mu \to E \otimes_A F

for every ordered pair $(\lambda, \mu)$ , and we take $h_{\lambda \mu} = i_\lambda \otimes j_\mu$ , where $i_\lambda : E_\lambda \to E$ and $j_\mu : F_\mu \to F$ are the canonical injections. Then clearly $h \circ g$ coincides with the identity mapping for the elements of the form $\left( \sum_\lambda x_\lambda \right) \otimes \left( \sum_\mu y_\mu \right)$ which generate the $\mathbf{Z}$ -module $E \otimes_A F$ ; similarly $g \circ h$ coincides with the identity mapping for the elements of the form $\sum_{\lambda, \mu} (x_\lambda \otimes y_\mu)$ which generate the $\mathbf{Z}$ -module $G$ , since for each ordered pair $(\lambda, \mu)$ the products

x_\lambda \otimes y_\mu \quad (x_\lambda \in E_\lambda, y_\mu \in F_\mu)

generate the $\mathbf{Z}$ -module $E_\lambda \otimes_A F_\mu$ . Whence the proposition.

Let $u_\lambda : E_\lambda \to E'_\lambda$ , $v_\mu : F_\mu \to F'_\mu$ be $A$ -homomorphisms; clearly the diagram

\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}

is commutative.

Corollary 1

If the left $A$ -module $F$ admits a basis $(b_\mu)_{\mu \in M}$ , every element of $E \otimes_A F$ can be written uniquely in the form $\sum_\mu (x_\mu \otimes b_\mu)$ , where $x_\mu \in E$ and the family $(x_\mu)$ has finite support. The $\mathbf{Z}$ -module $E \otimes_A F$ is isomorphic to $E^{(M)}$ considered as a $\mathbf{Z}$ -module.

Cites 2 · Cited by 9

Cites Proposition 4, Proposition 7

Cited by Lemma 10, AC I § 2, Corollary 3, Corollary 4, Lemma 3, II § 4, Proposition 4, II § 5, Corollary, II § 7, Proposition 6, III § 4, Proposition 7, III § 4, Proposition 5, VIII § 5

The basis $(b_\mu)$ defines an isomorphism of $F$ onto $\bigoplus_{\mu \in M} Ab_\mu$ , whence there is an isomorphism $E \otimes_A F \to \bigoplus_{\mu \in M} (E \otimes_A Ab_\mu)$ by virtue of Proposition 7; as ξ ↦ ξb_μ is an isomorphism of A_s onto Ab_μ, x ↦ x ⊗ b_μ is an isomorphism of E onto E ⊗_A (Ab_μ) by virtue of Proposition 4 of no. 4, whence the corollary.

If E is a ((B'_i); A, (C'_j))-multimodule, the canonical isomorphism E ⊗_A F → E^{(M)} is a ((B'_i); (C'_j))-multimodule isomorphism.

In particular, if also E admits a basis (a_λ)_{λ ∈ L}, every z ∈ E ⊗_A F may be written in one and only one way in the form $\sum_{\lambda, \mu} (a_\lambda \xi_{\lambda\mu}) \otimes b_\mu$ , where the ξ_{λμ} belong to A (and form a family of finite support); the mapping $z \mapsto (\xi_{\lambda\mu})_{(\lambda, \mu) \in L \times M}$ is an isomorphism of E ⊗_A F onto A^{(L × M)} for the \mathbf{Z}-module structures (and even the module structures over the centre of A). More particularly:

Corollary 2

If E and F are two free modules over a commutative ring C and (a_λ) (resp. (b_μ)) is a basis of the C-module E (resp. F), then (a_λ ⊗ b_μ) is a basis of the C-module E ⊗_C F.

Cited by 9

Cited by II § 10. Matrices, § 3. Tensor products, Corollary 7, Remark, II § 7, III § 1. Algebras, Proposition 4, III § 4, Remark, III § 5, Theorem 1, III § 7, Proposition 1, VIII § 12

By an abuse of language, the basis (a_λ ⊗ b_μ) is sometimes called the tensor product of the bases (a_λ) and (b_μ).

Remark (1). Let E be a free right A-module, F a free left A-module, (a_λ)_{λ ∈ L} a basis of E and (b_μ)_{μ ∈ M} a basis of F. Every element z ∈ E ⊗_A F may be written uniquely as $\sum_\lambda a_\lambda \otimes y_\lambda$ , where y_λ ∈ F, and also uniquely as $\sum_\mu x_\mu \otimes b_\mu$ , where x_μ ∈ E. If we write $y_\lambda = \sum_\mu \eta_{\lambda\mu} b_\mu$ , $x_\mu = \sum_\lambda a_\lambda \xi_{\lambda\mu}$ , where the ξ_{λμ} and η_{λμ} belong to A, then $\xi_{\lambda\mu} = \eta_{\lambda\mu}$ for all (λ, μ), for

\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).

Corollary 3

Let (E_λ)_{λ ∈ L} be a family of right A-modules and F a free (resp. finitely generated free) left A-module. Then the canonical mapping (22)

Cites 3 · Cited by 2

Cites Corollary 1, Proposition 7, Proposition 4

Cited by § 3. Tensor products, Proposition 15, II § 7

\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_A F \to \prod_{\lambda \in L} (E_\lambda \otimes_A F)

is injective (resp. bijective).

If (b_μ) is a basis of F, every element of $\left( \prod_{\lambda \in L} E_\lambda \right) \otimes_A F$ can be written uniquely as $z = \sum_\mu ((x_\lambda^{(\mu)}) \otimes b_\mu)$ (Corollary 1); to say that its canonical image is zero means that, for all λ ∈ L, $\sum_\mu (x_\lambda^{(\mu)} \otimes b_\mu) = 0$ , hence $x_\lambda^{(\mu)} = 0$ for all λ ∈ L and all μ (Corollary 1) and therefore $z = 0$ .

Showing that the canonical mapping is bijective when F admits a finite basis is immediately reduced, by virtue of Proposition 7, to the case where

F = A_s; but then the two sides are canonically identified with $\prod_{\lambda \in L} E_\lambda$ (no. 4, Proposition 4) and after these identifications the canonical mapping (22) becomes the identity.

Corollary 4

Let A be a ring with no divisor of zero, E a free right A-module and F a free left A-module. Then the relation x $\otimes$ y = 0 in E $\otimes_A$ F implies x = 0 or y = 0.

Cites 1 · Cited by 1

Cites Corollary 1

Cited by Proposition 1, VII § 4

Let $(a_\lambda)$ be a basis of E, $(b_\mu)$ a basis of F and let $x = \sum_\lambda a_\lambda \xi_\lambda, y = \sum_\mu \eta_\mu b_\mu$ ; then $x \otimes y = \sum_{\lambda, \mu} ((a_\lambda \xi_\lambda \eta_\mu) \otimes b_\mu$ and the relation $x \otimes y = 0$ implies $\xi_\lambda \eta_\mu = 0$ for every ordered pairs of indices $(\lambda, \mu)$ (Corollary 1). Hence, if $x \neq 0$ , that is $\xi_\lambda \neq 0$ for at least one $\lambda$ , it follows that $\eta_\mu = 0$ for all $\mu$ , whence $y = 0$ .

Corollary 5

Let E be a right A-module, F a left A-module, M a submodule of E and N a submodule of F. If M is a direct factor of E and N a direct factor of F, the canonical homomorphism $M \otimes_A N \to E \otimes_A F$ is injective and the image of $M \otimes_A N$ under this homomorphism is a direct factor of the $\mathbf{Z}$ -module $E \otimes_A F$ .

Cites 1 · Cited by 4

Cites Proposition 7

Cited by Lemma 1, AC I § 2, Corollary 7, Proposition 14, II § 7, Proposition 2, III § 11

This follows immediately from Proposition 7.

Note that if E is a $((B'_i); A, (C'_j))$ -multimodule and F an $(A, (B''_h); (C''_k))$ -multimodule and M and N direct factors in these multimodules, $M \otimes N$ is a direct factor of the $((B'_i), (B''_h); (C'_j), (C''_k))$ -multimodule $E \otimes F$ .

Corollary 6

Let P be a projective left A-module and E, F two right A-modules. For every injective homomorphism $u : E \to F$ , the homomorphism*

u \otimes 1_P : E \otimes_A P \to F \otimes_A P

*is injective.

Cites 4 · Cited by 4

Cites Proposition 4, II § 2, Proposition 7, Corollary 1, II § 1, Proposition 4

Cited by Proposition 7, II § 11, Corollary, II § 4, Proposition 9, VIII § 11, Proposition 11, VIII § 6

There exists a left A-module Q such that $L = P \oplus Q$ is free (§ 2, no. 2, Proposition 4) and $u \otimes 1_L$ is identified (Proposition 7) with

(u \otimes 1_P) \oplus (u \otimes 1_Q);

it therefore suffices to prove the corollary when P is free (§ 1, no. 6, Corollary 1 to Proposition 7). The same argument reduces the problem to the case where

P = A_s

, which follows immediately from no. 4, Proposition 4.

Corollary 7

Let C be a commutative ring. If E and F are two projective C-modules, $E \otimes_C F$ is a projective C-module.

Cites 2

Cites Corollary 2, Corollary 5

This follows immediately from Corollary 5 and the fact that the tensor product of two free C-modules is a free C-module (Corollary 2).

Remark 2

Under the hypotheses of Proposition 7, let $E'_\lambda$ be a submodule of $E_\lambda$ , $F'_\mu$ a submodule of $F_\mu$ and let $E' = \bigoplus_{\lambda \in L} E'_\lambda$ , $F' = \bigoplus_{\mu \in M} F'_\mu$ . Let Im $(E' \otimes_A F')$ (resp. Im(E'_λ ⊗_A F'_μ)) denote the image of E' ⊗_A F' (resp. E'_λ ⊗_A F'_μ) in E ⊗_A F (resp. E_λ ⊗_A F_μ) under the canonical mapping; then the isomorphism (23) identifies the sub- $\mathbf{Z}$ -modules

\operatorname{Im}(E' \otimes_A F') \quad \text{and} \quad \bigoplus_{(\lambda, \mu) \in I \times M} \operatorname{Im}(E'_\lambda \otimes_A F'_\mu);

this follows immediately from the commutativity of the diagram

\begin{array}{ccc} \left( \bigoplus_\lambda E'_\lambda \right) \otimes_A \left( \bigoplus_\mu F'_\mu \right) & \longrightarrow & \left( \bigoplus_\lambda E_\lambda \right) \otimes_A \left( \bigoplus_\mu F_\mu \right) \\ \downarrow & & \downarrow \\ \bigoplus_{\lambda, \mu} (E'_\lambda \otimes_A F'_\mu) & \longrightarrow & \bigoplus_{\lambda, \mu} (E_\lambda \otimes_A F_\mu) \end{array}

where the vertical arrows are the canonical isomorphisms.

Cites 1

Cites Proposition 7

8. Associativity of the tensor product

Proposition 8

Let A, B be two rings, E a right A-module, F an (A, B)-bimodule and G a left B-module. Then E ⊗_A F is a right B-module, F ⊗_B G a left A-module and there exists one and only one $\mathbf{Z}$ -linear mapping

Cites 1 · Cited by 6

Cites Proposition 1

Cited by Proposition 8, AC I § 2, § 3. Tensor products, Proposition 3, II § 5, Proposition 6, II § 5, Example, III § 4, Definition 1, VIII § 6

\phi : (E \otimes_A F) \otimes_B G \to E \otimes_A (F \otimes_B G)

such that $\phi((x \otimes y) \otimes z) = x \otimes (y \otimes z)$ for $x \in E,\ y \in F,\ z \in G;$ moreover this $\mathbf{Z}$ -linear mapping is bijective ("associativity" of the tensor product).

The right B-module structure on E ⊗_A F and left A-module structure on F ⊗_B G have been defined in no. 4. The uniqueness of $\phi$ is obvious since the elements $(x \otimes y) \otimes z$ generate the $\mathbf{Z}$ -module $(E \otimes_A F) \otimes_B G$ . To show the existence of $\phi$ , we note that, for all $z \in G,\ h_z : y \mapsto y \otimes z$ is an A-linear mapping of the left A-module F into the left A-module F ⊗_B G. We write $g_z = 1_E \otimes h_z$ , which is therefore a $\mathbf{Z}$ -linear mapping of E ⊗_A F into E ⊗_A (F ⊗_B G) and consider the mapping $(t, z) \mapsto g_z(t)$ from $(E \otimes_A F \times G$ into $E \otimes_A (F \otimes_B G)$ ; as $h_{z + z'} = h_z + h_{z'}$ for $z \in G,\ z' \in G$ , it is immediate that the above mapping is $\mathbf{Z}$ -bilinear. Further, we show that for all $\mu \in B,\ g_{\mu z}(t) = g_z(t \mu)$ ; it is obviously sufficient to do this for $t = x \otimes y$ where $x \in E$ and $y \in F$ ; now and

g_{\mu z}(x \otimes y) = x \otimes (y \otimes \mu z)

g_z((x \otimes y)\mu) = g_z(x \otimes y\mu) = x \otimes (y\mu \otimes z).

Proposition 1 (no. 1) then proves the existence of a $\mathbf{Z}$ -linear mapping

\phi : (E \otimes_A F) \otimes_B G \to E \otimes_A (F \otimes_B G)

such that $\phi(t \otimes z) = g_z(t)$ , hence $\phi((x \otimes y) \otimes z) = x \otimes (y \otimes z)$ . Similarly a $\mathbf{Z}$ -linear mapping

\psi : E \otimes_A (F \otimes_B G) \to (E \otimes_A F) \otimes_B G

is defined such that $\psi(x \otimes (y \otimes z)) = (x \otimes y) \otimes z$ and clearly $\psi \circ \phi$ and $\phi \circ \psi$ are the identity mappings of $(E \otimes_A F) \otimes_B G$ and $E \otimes_A (F \otimes_B G)$ respectively, since they reduce to the identity mappings on generating systems of these $\mathbf{Z}$ -modules.

It is immediate that, if $E$ is a $((C'_i); A, (D'_j))$ -multimodule, $F$ an $(A, (C''_h); B, (D''_k))$ -multimodule and $G$ a $(B, (C'''_l); (D'''_m))$ -multimodule, the canonical isomorphism defined in Proposition 8 is a $((C'_i), (C''_h), (C'''_l); (D'_j), (D''_k), (D'''_m))$ -multimodule isomorphism. In particular, if $C$ is a commutative ring and $E, F, G$ three $C$ -modules, there is a canonical $C$ -module isomorphism

(E \otimes_C F) \otimes_C G \to E \otimes_C (F \otimes_C G).

We shall see below that, under certain conditions, the definition of tensor product can be generalized to a family of multimodules, which will in particular give us under the hypotheses of Proposition 8 a $\mathbf{Z}$ -module $E \otimes_A F \otimes_B G$ , which is canonically isomorphic to each of the $\mathbf{Z}$ -modules $(E \otimes_A F) \otimes_B G$ and $E \otimes_A (F \otimes_B G)$ and with which the latter will be identified.

9. Tensor product of families of multimodules

Let $(G_\lambda)_{\lambda \in L}$ be a family of $\mathbf{Z}$ -modules; a mapping $u$ of the set $G = \prod_{\lambda \in L} G_\lambda$ into a $\mathbf{Z}$ -module is said to be multiadditive (or $\mathbf{Z}$ -*multilinear*) if $(x_\lambda) \mapsto u((x_\lambda))$ is additive with respect to each of the variables $x_\lambda$ ; to be precise, this means that, for all $\mu \in L$ and every element $(a_\lambda) \in \prod_{\lambda \neq \mu} G_\lambda$ , canonically identifying $G$ with $G_\mu \times \prod_{\lambda \neq \mu} G_\lambda$ ,

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.

This implies in particular that $u((x_\lambda)) = 0$ if one of the $x_\lambda$ is zero.

We consider also the universal mapping problem where $\Sigma$ is the species of $\mathbf{Z}$ -module structure and the $\alpha$ -mappings are the multiadditive mappings of $G$ into a $\mathbf{Z}$ -module. A solution is still obtained by considering the $\mathbf{Z}$ -module $C = \mathbf{Z}^{(G)}$ of formal linear combinations of elements of $G$ with coefficients in $\mathbf{Z}$ and the sub- $\mathbf{Z}$ -module $D$ of $C$ generated by the elements of the form

(x_\mu + y_\mu, (z_\lambda)_{\lambda \neq \mu}) - (x_\mu, (z_\lambda)_{\lambda \neq \mu}) - (y_\mu, (z_\lambda)_{\lambda \neq \mu})

where $\mu \in L, x_\mu \in G_\mu, y_\mu \in G_\mu$ and the $z_\lambda \in G_\lambda$ ( $\lambda \neq \mu$ ) are arbitrary. The quotient $\mathbf{Z}$ -module $C/D$ is called the tensor product (over $\mathbf{Z}$ ) of the family $(G_\lambda)_{\lambda \in L}$ of $\mathbf{Z}$ -*modules* and is denoted by $\bigotimes_{\lambda \in L} G_\lambda$ ; for every element $(x_\lambda)_{\lambda \in L}$ of $G$ which is an element of the canonical basis of $C$ , $\bigotimes_{\lambda \in L} x_\lambda$ denotes the canonical image of this element in C/D. It follows from the above definitions that the mapping $\phi : (x_\lambda) \mapsto \bigotimes_{\lambda \in L} x_\lambda$ of G into $\bigotimes_{\lambda \in L} G_\lambda$ is $\mathbf{Z}$ -multilinear and that, for every $\mathbf{Z}$ -multilinear mapping $f$ of G into a $\mathbf{Z}$ -module H, there exists one and only one $\mathbf{Z}$ -linear mapping $g : \bigotimes_{\lambda \in L} G_\lambda \to H$ such that $f = g \circ \phi$ ; the ordered pair $\left( \bigotimes_{\lambda \in L} G_\lambda, \phi \right)$ thus resolves the universal mapping problem in question.

Let $(G'_\lambda)_{\lambda \in L}$ be another family of $\mathbf{Z}$ -modules and, for all $\lambda \in L$ , let $v_\lambda : G_\lambda \to G'_\lambda$ be a $\mathbf{Z}$ -linear mapping (in other words a homomorphism of commutative groups). Then the mapping

(x_\lambda) \mapsto \bigotimes_{\lambda \in L} v_\lambda(x_\lambda)

of G into

\bigotimes_{\lambda \in L} G'_\lambda

\mathbf{Z}

-multilinear and hence defines canonically a

\mathbf{Z}

-linear mapping of

\bigotimes_{\lambda \in L} G_\lambda

into

\bigotimes_{\lambda \in L} G'_\lambda

denoted by

\bigotimes_{\lambda \in L} v_\lambda

and such that

\left( \bigotimes_{\lambda \in L} v_\lambda \right) \left( \bigotimes_{\lambda \in L} x_\lambda \right) = \bigotimes_{\lambda \in L} v_\lambda(x_\lambda).

In particular, we consider, for some

\mu \in L

, an endomorphism

\theta

G_\mu

; we denote by

\tilde{\theta}

the endomorphism of

\bigotimes_{\lambda \in L} G_\lambda

equal to

\bigotimes_{\lambda \in L} v_\lambda

where

v_\mu = \theta

and

v_\lambda = 1_{G_\lambda}

for

\lambda \neq \mu

Then we suppose given a set $\Omega$ , a mapping

c : \omega \mapsto (\rho(\omega), \sigma(\omega))

\Omega

into

L \times L

and, for all

\omega \in \Omega

, an endomorphism

p_\omega

G_{\rho(\omega)}

and an endomorphism

q_\omega

G_{\sigma(\omega)}

; there correspond to them two endomorphisms

\tilde{p}_\omega

and

\tilde{q}_\omega

P = \bigotimes_{\lambda \in L} G_\lambda

. Let R be the sub-

\mathbf{Z}

-module of P generated by the union of the images of the endomorphisms

\tilde{p}_\omega - \tilde{q}_\omega

when

\omega

runs through

\Omega

. The quotient

\mathbf{Z}

-module

P/R

is called the tensor product of the family

(G_\lambda)_{\lambda \in L}

relative to c, p, q and is denoted by

\bigotimes_{(c, p, q)} G_\lambda

; composing the canonical homomorphism

P \to P/R

with the mapping

\phi : G \to \bigotimes_{\lambda \in L} G_\lambda

defined above, we obtain a

\mathbf{Z}

-multilinear mapping

\phi_{(c, p, q)} : G \to \bigotimes_{(c, p, q)} G_\lambda

and write

\phi_{(c, p, q)}((x_\lambda)) = \bigotimes_{(c, p, q)} x_\lambda

or simply

\bigotimes_{(c)} x_\lambda

. The ordered pair consisting of

\bigotimes_{(c, p, q)} x_\lambda

and

\phi_{(c, p, q)}

resolves the following universal mapping problem: let

\bar{p}_\omega

(resp.

\bar{q}_\omega

) denote the mapping

(x_{\rho(\omega)}, (x_\lambda)_{\lambda \neq \rho(\omega)}) \mapsto (p_\omega(x_{\rho(\omega)}), (x_\lambda)_{\lambda \neq \rho(\omega)})

(resp.

(x_{\sigma(\omega)}, (x_\lambda)_{\lambda \neq \sigma(\omega)}) \mapsto (q_\omega(x_{\sigma(\omega)}), (x_\lambda)_{\lambda \neq \sigma(\omega)})

) of G into itself. Then

\Sigma

is taken to be the species of

\mathbf{Z}

-module structure and the α-mappings the

\mathbf{Z}

-multilinear mappings

u

G

into a

\mathbf{Z}

-module which further satisfy the conditions

(26)

u \circ \bar{p}_\omega = u \circ \bar{q}_\omega

for all $\omega \in \Omega$ . The proof is obvious from the above definitions.

This construction recovers in particular that of $E \otimes_A F$ described in no. 1: in this case we take $L = \{1, 2\}$ , $G_1 = E$ , $G_2 = F$ , $\Omega = A$ ; further, for all $\omega \in A$ , we must have $\rho(\omega) = 1$ , $\sigma(\omega) = 2$ , $\omega$ is the endomorphism $x \mapsto x\omega$ of the $\mathbf{Z}$ -module $E$ and $q_\omega$ the endomorphism $y \mapsto \omega y$ of the $\mathbf{Z}$ -module $F$ .

Let $(G'_\lambda)_{\lambda \in L}$ be a second family of $\mathbf{Z}$ -modules; keeping the mapping $c$ the same, suppose given for all $\omega \in \Omega$ , an endomorphism $p'_\omega$ of $G'_{\rho(\omega)}$ and an endomorphism $q'_\omega$ of $G'_{\sigma(\omega)}$ . For all $\lambda \in L$ , then let $v_\lambda : G_\lambda \to G'_\lambda$ be a $\mathbf{Z}$ -linear mapping such that, for all $\omega \in \Omega$ ,

(27)

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)}

(in other words, for all $\lambda \in L$ , $v_\lambda$ is a morphism for the laws of action on $G_\lambda$ (resp. $G'_\lambda$ ) defined on the $p_\xi$ and $q_\eta$ (resp. $p'_\xi$ and $q'_\eta$ ) with $\xi$ and $\eta$ such that $\rho(\xi) = \lambda$ and $\sigma(\eta) = \lambda$ ). Then the mapping

u : (x_\lambda) \mapsto \bigotimes_{(c, p', q')} v_\lambda(x_\lambda)

G

into

\bigotimes_{(c, p', q')} G'_\lambda

\mathbf{Z}

-multilinear and satisfies conditions (26) and hence defines a

\mathbf{Z}

-linear mapping of

\bigotimes_{(c, p, q)} G_\lambda

into

\bigotimes_{(c, p', q')} G'_\lambda

, which we shall denote simply by

\bigotimes_{(c)} v_\lambda

if no confusion can arise.

We shall now give an "associativity" property for the general tensor products thus defined. Let $(L_i)_{1 \leq i \leq n}$ be a finite partition of $L$ ; for every index $i$ , let $\Omega_i$ denote the subset of $\Omega$ consisting of the elements such that $\rho(\omega) \in L_i$ and $\sigma(\omega) \in L_i$ ; clearly the $\Omega_i$ are pairwise disjoint; we set $\Omega' = \Omega - \left( \bigcup_i \Omega_i \right)$ . For each index $i$ , $c^{(i)}$ will denote the mapping $\omega \mapsto (\rho(\omega), \sigma(\omega))$ of $\Omega_i$ into $L_i \times L_i$ ; for $\omega \in \Omega_i$ , we write $p^{(i)}_\omega$ and $q^{(i)}_\omega$ instead of $p_\omega$ and $q_\omega$ . Then for each $i$ there is a "partial" tensor product

F_i = \bigotimes_{(c^{(i)}, p^{(i)}, q^{(i)})} G_\lambda.

We shall further make the following "permutability" hypothesis:

(P) If $\omega \in \Omega'$ , $p_\omega$ (resp. $q_\omega$ ) permutes with each of the endomorphisms $p_\xi$ and $q_\eta$ of $G_{\rho(\omega)}$ (resp. $G_{\sigma(\omega)}$ ) such that $\xi \notin \Omega'$ , $\eta \notin \Omega'$ and $\rho(\omega) = \rho(\xi) = \sigma(\eta)$ (resp. $\sigma(\omega) = \rho(\xi) = \sigma(\eta)$ ).

For each $\omega \in \Omega'$ , let $i$ be the index such that $\rho(\omega) \in L_i$ ; then consider the family $(v_\lambda)_{\lambda \in L_i}$ where $v_{\rho(\omega)} = p_\omega$ and $v_\lambda = 1_{G_\lambda}$ for $\lambda \neq \rho(\omega)$ ; hypothesis (P) implies that the family $(v_\lambda)$ satisfies conditions (27) (where $p'$ and $p$ must be replaced by $p^{(i)}$ , $q'$ and $q$ by $q^{(i)}$ , $\omega$ by an element $\xi$ running through $\Omega_i$ ); thus an endomorphism $\bigotimes_{(c^{(i)})} v_\lambda = r_\omega$ is derived of the $\mathbf{Z}$ -module $F_i$ . Similarly, an endomorphism $s_\omega$ is defined of the $\mathbf{Z}$ -module $F_j$ starting with $q_\omega$ , $j$ being the index such that $\sigma(\omega) \in L_j$ ; finally let $d(\omega) = (i, j)$ . Then we can define the tensor product $\bigotimes_{(d, r, s)} F_i$ and the corresponding canonical mapping

\phi_{(d, r, s)} : \prod_{i=1}^n F_i \to \bigotimes_{(d, r, s)} F_i.

On the other hand, for each $i$ , the canonical mapping

\psi_i = \phi_{(c^{(i)}, p^{(i)}, q^{(i)})} : \prod_{\lambda \in L_i} G_\lambda \to F_i;

using the associativity of the product of sets, a $\mathbf{Z}$ -multilinear mapping $\psi = \phi_{(d, r, s)} \circ (\psi_i)$ of $G$ into $\bigotimes_{(d, r, s)} F_i$ is derived. We show that the ordered pair $\left( \bigotimes_{(d, r, s)} F_i, \psi \right)$ is a solution of the same universal problem as $\left( \bigotimes_{(c, p, q)} G_\lambda, \phi_{(c, p, q)} \right)$ , whence will follow the existence of a unique $\mathbf{Z}$ -module isomorphism

\theta : \bigotimes_{(c, p, q)} G_\lambda \to \bigotimes_{(d, r, s)} F_i

such that $\psi = \theta \circ \phi_{(c, p, q)}$ (Set Theory, IV, § 3, no. 1).

By induction on $n$ , the proof is reduced to the case $n = 2$ ; for simplicity we write $F_1 \otimes_{(d)} F_2$ and $y_1 \otimes_{(d)} y_2$ instead of $\bigotimes_{(d, r, s)} F_i$ and $\bigotimes_{(d, r, s)} y_i$ . Consider the mapping from $G$ to $F_1 \otimes_{(d)} F_2$

h : (x_\lambda) \to \left( \bigotimes_{(c^{(1)})} x_\lambda \right) \otimes_{(d)} \left( \bigotimes_{(c^{(2)})} x_\lambda \right).

It is obviously $\mathbf{Z}$ -multilinear; we show that it satisfies conditions (26) for all $\omega \in \Omega$ . This is obvious if $\omega \in \Omega_1$ or $\omega \in \Omega_2$ ; otherwise, supposing, in order to fix the ideas, that $\rho(\omega) \in L_1$ and $\sigma(\omega) \in L_2$ , the values of $h \circ \bar{p}_\omega$ and $h \circ \bar{q}_\omega$ at $(x_\lambda)$ are respectively

\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)

which are also equal by definition of $F_1 \otimes_{(d)} F_2$ .

This being so, let $u$ be a $\mathbf{Z}$ -multilinear mapping of $G$ into a $\mathbf{Z}$ -module $H$ , satisfying conditions (26); we shall define a $\mathbf{Z}$ -linear mapping $v : F_1 \otimes_{(d)} F_2 \to H$ such that $u = v \circ h$ and that will prove our assertion (repeating the argument of no. 1, Corollary 1 to Proposition 1). For all $z_2 = (x_\lambda)_{\lambda \in L_2}$ we consider the "partial" linear mapping of $\prod_{\lambda \in L_1} G_\lambda$ into $H$

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}).

Clearly it is $\mathbf{Z}$ -multilinear and satisfies conditions (26) for $\omega \in \Omega_1$ ; by definition there thus exists a $\mathbf{Z}$ -linear mapping $y_1 \mapsto w_1(y_1, x_2)$ of $F_1$ into $H$ such that

w_1\left( \bigotimes_{(c^{(1)})} x_\lambda, z_2 \right) = u((x_\lambda)_{\lambda \in L_1}, z_2)

We next consider the mapping

u_2: (x_\lambda)_{\lambda \in L_2} \mapsto w_1(., (x_\lambda)_{\lambda \in L_2})

of $\prod_{\lambda \in L_2} G_\lambda$ into $\mathrm{Hom}_\mathbf{Z}(F_1, H)$ ; it is obviously $\mathbf{Z}$ -multilinear and satisfies conditions (26) for $\omega \in \Omega_2$ , by virtue of the hypothesis on $u$ and relations (28) and (29) and taking account of the fact that the elements of the form $\bigotimes_{(c^{(1)})} x_\lambda$ generate the $\mathbf{Z}$ -module $F_1$ . Hence there is a $\mathbf{Z}$ -linear mapping

w_2: F_2 \to \mathrm{Hom}_\mathbf{Z}(F_1, H)

such that

w_2\left( \bigotimes_{(c^{(2)})} x_\lambda \right) = u_2((x_\lambda)_{\lambda \in L_2})

or also

\left( w_2\left( \bigotimes_{(c^{(2)})} x_\lambda \right) \right)\left( \bigotimes_{(c^{(1)})} x_\lambda \right) = u((x_\lambda)_{\lambda \in L}).

We now consider, for $y_1 \in F_1, y_2 \in F_2$ , the element of $H$

w(y_1, y_2) = (w_2(y_2))(y_1).

Clearly $w$ is a $\mathbf{Z}$ -bilinear mapping of $F_1 \times F_2$ into $H$ . We show further that, for all $\omega \in \Omega'$ , (assuming, to fix the ideas, that $\rho(\omega) \in L_1$ and $\sigma(\omega) \in L_2$ )

w(r_\omega(y_1), y_2) = w(y_1, s_\omega(y_2)).

It suffices to verify this relation when $y_1$ (resp. $y_2$ ) is of the form $\bigotimes_{(c^{(1)})} x_\lambda$ (resp. $\bigotimes_{(c^{(2)})} x_\lambda$ ), since these elements generate the $\mathbf{Z}$ -module $F_1$ (resp. $F_2$ ). But by definition, $r_\omega\left( \bigotimes_{(c^{(1)})} x_\lambda \right) = \bigotimes_{(c^{(1)})} x'_\lambda$ , where $x'_{\rho(\omega)} = p_\omega(x_{\rho(\omega)})$ and $x'_\lambda = x_\lambda$ for λ ≠ ρ(ω) in L₁; similarly $s_\omega\left( \bigotimes_{(c^{(2)})} x_\lambda \right) = \bigotimes_{(c^{(2)})} x''_\lambda$ , where $x''_{\sigma(\omega)} = q_\omega(x'_{\sigma(\omega)})$ and $x''_\lambda = x_\lambda$ for λ ≠ σ(ω) in L₂; using (30) and (31), relation (32) then follows from (26). Hence there exists a $\mathbf{Z}$ -linear mapping v of F₁ ⊗ F₂ into H such that $v(y_1 \otimes y_2) = w(y_1, y_2)$ and it then follows from (30) and (31) that $v \circ h = u$ .

The most important special case of the general tensor product defined above is the following: we start with a family $(A_i)_{1 \leq i \leq n-1}$ of rings and a family $(E_i)_{1 \leq i \leq n}$ , where $E_1$ is a right $A_1$ -module, $E_n$ is a left $A_{n-1}$ module and for $2 \leq i \leq n-1$ , $E_i$ is an $(A_{i-1}, A_i)$ -bimodule. Then the above definition is applied as follows: L is the set $[1, n]$ , $G_i = E_i$ , Ω is the set the sum of the $A_i$ ( $1 \leq i \leq n-1$ ). For $\omega \in A_i$ ( $1 \leq i \leq n-1$ ), take $\rho(\omega) = i$ , $\sigma(\omega) = i + 1$ , $p_\omega$ is the endomorphism $x \mapsto x\omega$ of the $Z$ -module $E_i$ and $q_\omega$ the endomorphism $y \mapsto \omega y$ of the $Z$ -module $E_{i+1}$ ; the corresponding tensor product is denoted by

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

(a notation where the $A_i$ may occasionally be suppressed) and the elements $\bigotimes_{(c, p, q)} x_i$ of this tensor product, for a family $(x_i)$ such that $x_i \in E_i$ for $1 \leq i \leq n$ , may be written $x_1 \otimes x_2 \otimes \cdots \otimes x_n$ if no confusion can arise; an analogous notation is used for a $Z$ -linear mapping $\bigotimes_{(c)} v_i$ . Hypothesis (P) holds for every partition of $[1, n]$ , since the $E_i$ are bimodules for $2 \leq i \leq n-1$ . When $n = 3$ , we have thus defined the $Z$ -module $E \otimes_A F \otimes_B G$ alluded to in no. 8, and recovered Proposition 8 (no. 8).

When each of the $E_i$ is a multimodule (with, for $2 \leq i \leq n-1$ , $A_{i-1}$ one of the rings operating on the left and $A_i$ one of the rings operating on the right and analogous conditions for $i = 1$ and $i = n$ ), as in no. 4, a multimodule structure is defined on $E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n$ with respect to all the rings except the $A_i$ which operate on the $E_i$ ( $1 \leq i \leq n$ ).

In particular, let C be a commutative ring, $(E_i)_{1 \leq i \leq n}$ a family of C-modules. By giving $E_1$ and $E_n$ two C-module structures identical with the given structure and $E_i$ for $2 \leq i \leq n-1$ three C-module structures identical with the given structure, we define on the tensor product

E_1 \otimes_C E_2 \otimes_C E_3 \otimes \cdots \otimes_C E_{n-1} \otimes_C E_n

$n$ C-modules structures which are compatible with one another and which are in fact identical, since for all $\gamma \in C$ and $(x_i) \in \prod_{i=1}^n E_i$ , by definition

(\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).

When we speak of the tensor product (34) as a C-module, we shall always mean with this structure, unless otherwise mentioned, and the tensor product (34) is also denoted by $\bigotimes_{1 \leq i \leq n} E_i$ if no confusion can arise. For every C-module G, the $\mathbf{Z}$ -multilinear mappings of $\prod_{i=1}^n E_i$ into G which, for every index i, satisfy the relation

f(x_1, \ldots, x_{i-1}, \gamma x_i, x_{i+1}, \ldots, x_n) = \gamma f(x_1, \ldots, x_n)

for $\gamma \in \mathbf{C}$ and $(x_i) \in \prod_i E_i$ are then called C-multilinear and form a C-module denoted by $\mathscr{L}_n(E_1, \ldots, E_n; G)$ ; the universal property of the tensor product (34) then allows us to define a canonical C-module isomorphism

\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)

which associates with every C-multilinear mapping f the C-linear mapping g such that

f(x_1, \ldots, x_n) = g(x_1 \otimes x_2 \otimes \cdots \otimes x_n).

A C-multilinear mapping of $E_1 \times \cdots \times E_n$ into $\mathbf{C}$ is also called an $n$ -linear form.

Let $(F_i)_{1 \leq i \leq n}$ be another family of C-modules; for every system of n C-linear mappings $u_i : E_i \to F_i, u_1 \otimes u_2 \otimes \cdots \otimes u_n$ (also denoted by $\bigotimes_{1 \leq i \leq n} u_i$ ) is a C-linear mapping of

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.

Further, $(u_1, \ldots, u_n) \mapsto u_1 \otimes u_2 \otimes \cdots \otimes u_n$ is a C-multilinear mapping of $\prod_i \operatorname{Hom}_\mathbf{C}(E_i, F_i)$ into

\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).

Hence there corresponds canonically to the latter mapping a C-linear mapping called canonical

\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)

\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)

generalizing that defined in no. 5 for $n = 2$ .

The general associativity property seen earlier can be specialized here as follows. Given a partition $(J_k)_{1 \leq k \leq m}$ of the interval $\{1, n\}$ of $\mathbf{N}$ , let, for each $k$ , $F_k$ be the tensor product $E_{i_1} \otimes_\mathbf{C} E_{i_2} \otimes \cdots \otimes_\mathbf{C} E_{i_r}$ , where $(i_1, \ldots, i_r)$ is the strictly increasing sequence of elements of $J_k$ . Then there is a canonical C-module isomorphism (called "associativity isomorphism")

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

which, in the above notation, maps the tensor product

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},

to the tensor product $x_1 \otimes x_2 \otimes \cdots \otimes x_n$ (where $x_i \in E_i$ for all $i$ ).

In particular, if $\pi$ is a permutation of $\{1, n\}$ , writing $J_k = \{\pi(k)\}$ for $1 \leq k \leq n$ , we obtain a canonical ("commutativity") isomorphism

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

which maps $x_{\pi(1)} \otimes x_{\pi(2)} \otimes \cdots \otimes x_{\pi(n)}$ to $x_1 \otimes x_2 \otimes \cdots \otimes x_n$ . We shall often identify the various tensor products which correspond to one another under these canonical isomorphisms.

For $1 \leq i \leq n$ , suppose that $E_i$ admits a basis $(b^{(i)}_{\lambda_i})_{\lambda_i \in L_i}$ ; by induction on $n$ , it follows from no. 7, Corollary 2 to Proposition 7 that the family $(b^{(1)}_{\lambda_1} \otimes b^{(2)}_{\lambda_2} \otimes \cdots \otimes b^{(n)}_{\lambda_n})$ , where $(\lambda_1, \ldots, \lambda_n)$ runs through $\prod_{1 \leq i \leq n} L_i$ , is a basis of $\bigoplus_{1 \leq i \leq n} E_i$ , sometimes called the tensor product of the bases $(b^{(1)}_{\lambda_1})$ in question.

Remark

(1) The above remarks concerning the case of modules over a commutative ring generalize as in no. 5, Remark 2 when there is a tensor product $E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n$ where the rings $A_i$ are not necessarily commutative and where, for each $i$ , there is a homomorphism $\rho_i : C \to A$ of the same commutative ring $C$ such that: (1) $\rho_i(C)$ is contained in the centre of $A_i$ ; (2) for $2 \leq i \leq n - 1$ , the $C$ -module structures on $E_i$ obtained using the homomorphisms $\rho_{i-1}$ and $\rho_i$ coincide. Then a $C$ -*module* structure is obtained on $E_1 \otimes_{A_1} E_2 \otimes \cdots \otimes_{A_{n-1}} E_n$ and canonical mappings analogous to (13) (no. 5), which will be left to the reader to describe.

(2) Let $A, B$ be two rings, $E$ a right $A$ -module, $E'$ a left $A$ -module, $F$ a right $B$ -module and $F'$ a left $B$ -module. The $\mathbf{Z}\text{-bilinear}$ mappings of $(E \otimes_A E') \times (F \otimes_B F')$ into a $\mathbf{Z}\text{-module}$ $G$ are then in one-to-one correspondence with the $\mathbf{Z}\text{-multilinear}$ mappings $f$ of $E \times E' \times F \times F'$ into $G$ satisfying the conditions

\begin{cases} f(x \lambda, x', y, y') = f(x, \lambda x', y, y') \\ f(x, x', y \mu, y') = f(x, x', y, \mu y') \end{cases}

for $\lambda \in A, \mu \in B, x \in E, x' \in E', y \in F, y' \in F'$ . The general constructions given in this no. reduce the proof of this to defining a canonical $\mathbf{Z}\text{-module}$ isomorphism between $(E \otimes_A E') \otimes_{\mathbf{Z}} (F \otimes_B F')$ and $E \otimes_A E' \otimes_{\mathbf{Z}} F \otimes_B F'$ , which follows from the associativity property of tensor products of the form (33).

Exercises

See the exercises for § 3.