§ 6. Inverse and direct limits of modules

Throughout this paragraph, I will denote a non-empty preordered set and $\alpha \leq \beta$ the preorder relation on I. Unless otherwise mentioned, the inverse and direct systems have indexing set I.

1. Inverse limits of modules

Let $(A_\alpha, \phi_{\alpha \beta})$ be an inverse system of rings (I, § 10, no. 1), $(E_\alpha, f_{\alpha \beta})$ an inverse system of commutative groups (written additively) (I, § 10, no. 1) and suppose that each $E_\alpha$ has a left $A_\alpha$ -module structure; moreover suppose that for $\alpha \leq \beta$ $(f_{\alpha \beta}, \phi_{\alpha \beta})$ is a dimorphism of $E_\beta$ into $E_\alpha$ (§ 1, no. 13), in other words that

f_{\alpha \beta}(\lambda_\beta x_\beta) = \phi_{\alpha \beta}(\lambda_\beta) f_{\alpha \beta}(x_\beta),

for

x_\beta \in E_\beta

\lambda_\beta \in A_\beta

; then it follows from I, § 10, no. 2 that

E = \lim \leftarrow E_\alpha

has a left module structure over

A = \lim \leftarrow A_\alpha

. For all

\alpha \in I

, let

f : E \to E_\alpha

\phi_\alpha : A \to A_\alpha

be the canonical mappings; then

(f_\alpha, \phi_\alpha)

is a dimorphism of E into

E_\alpha

. We shall say that

(E_\alpha, f_{\alpha\beta})

is an inverse system of left $A_\alpha$ -modules and that the

A

-module

E

is its inverse limit.

Let $(E'_\alpha, f'_{\alpha\beta})$ be another inverse system of left $A_\alpha$ -modules and, for all $\alpha$ , let $u_\alpha : E'_\alpha \to E_\alpha$ be an $A_\alpha$ -linear mapping, these mappings forming an inverse system; then $u = \lim u_\alpha$ is an $A$ -linear mapping of $\lim E'_\alpha$ into $\lim E_\alpha$ .

Moreover:

Proposition 1

Let* $(E_\alpha, f_{\alpha\beta}),\ (E'_\alpha, f'_{\alpha\beta}),\ (E''_\alpha, f''_{\alpha\beta})$ be three inverse systems of $A_\alpha$ -modules and $(u_\alpha),\ (v_\alpha)$ *two inverse systems of $A_\alpha$ -linear mappings such that the sequences

0 \longrightarrow E'_\alpha \xrightarrow{u_\alpha} E_\alpha \xrightarrow{v_\alpha} E''_\alpha

are exact for all $\alpha$ . Then, writing $u = \lim u_\alpha,\ v = \lim v_\alpha$ , the sequence

0 \longrightarrow \lim E'_\alpha \xrightarrow{u} \lim E_\alpha \xrightarrow{v} \lim E''_\alpha

is exact.

As $u_\alpha^{-1}(0) = \{0\}$ for all $\alpha$ , it follows from Set Theory, III, § 7, no. 2, Proposition 2 that $u^{-1}(0) = \{0\}$ , hence $u$ is injective; further, the $u_\alpha(E'_\alpha)$ form an inverse system of subsets of the $E_\alpha$ and thus $u(\lim E'_\alpha) = \lim u_\alpha(E'_\alpha)$ . As $u_\alpha(E'_\alpha) = v_\alpha^{-1}(0)$ by hypothesis, $v^{-1}(0) = \lim u_\alpha(E'_\alpha) = u(\lim E'_\alpha)$ (Set Theory, III, § 7, no. 2, Proposition 2), which completes the proof.

Remark

(1) Proposition 1 and its proof are valid for arbitrary groups, except for change of notation.

(2) Note that if there are exact sequences

0 \longrightarrow E'_\alpha \xrightarrow{u_\alpha} E_\alpha \xrightarrow{v_\alpha} E''_\alpha \longrightarrow 0

it does not necessarily follow* that the sequence

0 \longrightarrow \lim E'_\alpha \xrightarrow{u} \lim E_\alpha \xrightarrow{v} \lim E''_\alpha \longrightarrow 0

is exact; in other words, the inverse limit of an inverse system of surjective linear mappings is not necessarily surjective* (cf. Exercise 1).

Suppose now that the $A_\alpha$ are equal to the same ring $A$ and the $\phi_{\alpha\beta}$ to $1_A$ ; then for every inverse system $(E_\alpha, f_{\alpha\beta})$ of $A$ -modules, $E = \lim E_\alpha$ is an $A$ -module. Let $F$ be an $A$ -module and, for all $\alpha$ , let $u_\alpha : F \to E_\alpha$ be an $A$ -linear mapping such that $(u_\alpha)$ is an inverse system of mappings; then $u = \lim u_\alpha$ is an $A$ -linear mapping of $F$ into $\lim E_\alpha$ . Conversely, for every $A$ -linear mapping $v : F \to \lim E_\alpha$ , the family of $v_\alpha = f_\alpha \circ v$ is an inverse system of $A$ -linear mappings such that $v = \lim_{\leftarrow} v_\alpha$ . We note on the other hand that for $\alpha \leq \beta$ the mapping

\operatorname{Hom}(1_F, f_{\alpha \beta}) = \bar{f}_{\alpha \beta}: \operatorname{Hom}_A(F, E_\beta) \to \operatorname{Hom}_A(F, E_\alpha)

is a $\mathbf{Z}$ -module homomorphism such that $(\operatorname{Hom}_A(F, E_\alpha), \bar{f}_{\alpha \beta})$ is an inverse system of $\mathbf{Z}$ -modules; as $\bar{f}_{\alpha \beta}(v_\beta) = f_{\alpha \beta} \circ v_\beta$ , the above remarks can therefore be expressed as follows:

Proposition 2

For every inverse system* $(E_\alpha, f_{\alpha \beta})$ of $A$ -modules and every $A$ -module $F$ , the canonical mapping $u \mapsto (f_\alpha \circ u)$ is a $\mathbf{Z}$ -module *isomorphism

(2)

l_F: \operatorname{Hom}_A(F, \lim_{\leftarrow} E_\alpha) \to \lim_{\leftarrow} \operatorname{Hom}_A(F, E_\alpha).

Corollary

For every $A$ *-module homomorphism $v: F \to F'$ , the

\bar{v}_\alpha = \operatorname{Hom}(v, 1_{E_\alpha}): \operatorname{Hom}(F', E_\alpha) \to \operatorname{Hom}(F, E_\alpha)

form an inverse system of $\mathbf{Z}$ -linear mappings and the diagram

(3)

\begin{array}{ccc} \operatorname{Hom}(F', \lim_{\leftarrow} E_\alpha) & \xrightarrow{l_{F'}} & \lim_{\leftarrow} \operatorname{Hom}(F', E_\alpha) \\ \downarrow \operatorname{Hom}(v, 1_E) & & \downarrow \lim_{\leftarrow} \bar{v}_\alpha \\ \operatorname{Hom}(F, \lim_{\leftarrow} E_\alpha) & \xrightarrow{l_F} & \lim_{\leftarrow} \operatorname{Hom}(F, E_\alpha) \end{array}

is commutative.

For all $u \in \operatorname{Hom}(F', \lim_{\leftarrow} E_\alpha)$ , $l_F(u \circ v) = (f_\alpha \circ u \circ v)$ by definition and the commutativity of diagram (3) follows immediately from the definitions.

2. Direct limits of modules

Henceforth I is assumed to be right directed.

Let $(A_\alpha, \phi_{\beta \alpha})$ be a direct system of rings (I, § 10, no. 3), $(E_\alpha, f_{\beta \alpha})$ a direct system of commutative groups (written additively) (I, § 10, no. 3) and suppose that each $E_\alpha$ has a left $A_\alpha$ -module structure; further, suppose that, for $\alpha \leq \beta$ , $(f_{\beta \alpha}, \phi_{\beta \alpha})$ is a dimorphism of $E_\alpha$ into $E_\beta$ (§ 1, no. 13), in other words that

(4)

f_{\beta \alpha}(\lambda_\alpha x_\alpha) = \phi_{\beta \alpha}(\lambda_\alpha) f_{\beta \alpha}(x_\alpha)

for $x_\alpha \in E_\alpha, \lambda_\alpha \in A_\alpha$ ; then $E = \lim_{\longrightarrow} E_\alpha$ has a left module structure over $A = \lim_{\longrightarrow} A_\alpha$ (I, § 10, no. 4). For all $\alpha \in I$ , let $f_\alpha: E_\alpha \to E, \phi_\alpha: A_\alpha \to A$ be the canonical mappings; then $(f_\alpha, \phi_\alpha)$ is a dimorphism of $E_\alpha$ into $E$ . We shall say that $(E_\alpha, f_{\beta \alpha})$ is a direct system of left $A_\alpha$ -modules and that the $A$ *-module $E$ is its direct limit.

Let $(E'_\alpha, f'_{\beta \alpha})$ be another direct system of left $A_\alpha$ *-modules and, for all $\alpha$ , let $u_\alpha : E'_\alpha \to E_\alpha$ be an $A_\alpha$ -linear mapping, these mappings forming a direct system; then $u = \lim \to u_\alpha$ is an $A$ -linear mapping of $\lim \to E'_\alpha$ into $\lim \to E_\alpha$ .

Moreover:

Proposition 3

Let* $(E_\alpha, f_{\beta \alpha}), (E'_\alpha, f'_{\beta \alpha}), (E''_\alpha, f''_{\beta \alpha})$ be three direct systems of $A_\alpha$ *-modules and* $(u_\alpha), (v_\alpha)$ two direct systems of $A_\alpha$ *-linear mappings such that the sequences

E'_\alpha \xrightarrow{u_\alpha} E_\alpha \xrightarrow{v_\alpha} E''_\alpha

are exact for all $\alpha$ . Then, writing $u = \lim \to u_\alpha, v = \lim \to v_\alpha$ , the sequence

\lim \to E'_\alpha \xrightarrow{u} \lim \to E_\alpha \xrightarrow{v} \lim \to E''_\alpha

is exact.

$u(\lim \to E'_\alpha) = \lim \to u_\alpha(E'_\alpha)$ and $v^{-1}(0) = \lim \to v_\alpha^{-1}(0)$ (Set Theory, III, § 7, no. 6, Corollary to Proposition 7).

Loosely speaking, Proposition 3 can also be expressed by saying that passing to the direct limit preserves exactness.

Proposition 4

Let* $(E_\alpha, f_{\beta \alpha})$ be a direct system of $A_\alpha$ *-modules, $E = \lim \to E_\alpha$ its direct limit and $\phi_\alpha : A_\alpha \to A$ and $f_\alpha : E_\alpha \to E$ the canonical mappings for all $\alpha \in I$ . If, for all $\alpha \in I$ , $S_\alpha$ is a generating system of $E_\alpha$ , then $S = \bigcup_{\alpha \in I} f_\alpha(S_\alpha)$ is a generating system of $E$ .

Every $x \in E$ is of the form $f_\alpha(x_\alpha)$ for some $\alpha \in I$ and some $x_\alpha \in E_\alpha$ and by hypothesis $x = \sum_i \lambda^{(i)}_\alpha y^{(i)}_\alpha$ , where $\lambda^{(i)}_\alpha \in A_\alpha$ and $y^{(i)}_\alpha \in S_\alpha$ ; writing $\lambda^{(i)} = \phi_\alpha(\lambda^{(i)}_\alpha)$ , $y^{(i)} = f_\alpha(y^{(i)}_\alpha)$ , we obtain $x = \sum_i \lambda^{(i)} y^{(i)}$ .

Proposition 5

With the hypotheses and notation of Proposition 4, suppose that for all* $\alpha \in I$ , $E_\alpha$ is the direct sum of a family $(M^\lambda_\alpha)_{\alpha \in L}$ of submodules (the indexing set $L$ being independent of $\alpha$ ) and that $f_{\beta \alpha}(M^\lambda_\alpha) \subset M^\lambda_\alpha$ for $\alpha \leq \beta$ and for all $\lambda \in L$ . Then $E$ is the direct sum of the submodules $M^\lambda = \lim \to M^\lambda_\alpha$ $(\lambda \in L)$ .

It follows from Proposition 4 that $E$ is the sum of the $M^\lambda$ . Let $(y^\lambda)_{\lambda \in L}$ be a family such that $y^\lambda \in M^\lambda$ for all $\lambda \in L$ and whose support is finite and suppose that $\sum_\lambda y^\lambda = 0$ . By virtue of Set Theory, III, § 7, no. 5, Lemma 1, there exist an $\alpha \in I$ and a family $(x^\lambda_\alpha)_{\alpha \in L}$ of finite support consisting of elements of $E_\alpha$ such that $x^\lambda_\alpha \in M^\lambda_\alpha$ and $y^\lambda = f_\alpha(x^\lambda_\alpha)$ for all $\lambda \in L$ . The relation $f_\alpha \left( \sum_{\lambda \in L} x^\lambda_\alpha \right) = 0$ implies the existence of a $\beta \geq \alpha$ such that $f_{\beta \alpha} \left( \sum_{\lambda \in L} x^\lambda_\alpha \right) = 0$ *(Set Theory, III, § 7, no. 5, Lemma 1), which may be written as $\sum_{\lambda \in L} x_\beta^\lambda = 0$ , where $x_\beta^\lambda = f_{\beta \alpha}(x_\alpha^\lambda) \in M_\beta^\lambda$ by hypothesis; hence $x_\beta^\lambda = 0$ for all $\lambda \in L$ and therefore $y^\lambda = f_\beta(x_\beta^\lambda) = 0$ for all $\lambda \in L$ , which proves that the sum of the $M^\lambda$ is direct.

Corollary

Let* $(P_\alpha)$ be a direct system of subsets of $E_\alpha$ and let $P = \lim \overrightarrow{P_\alpha}$ . If, for all $\alpha \in I$ , $P_\alpha$ is a free subset (resp. basis) of $E_\alpha$ , then $P$ is a free subset (resp. basis) of $E$ .

The second assertion follows immediately from the first and Proposition 4. It is therefore sufficient to prove that if the $P_\alpha$ are free every subset $\{ y^{(i)} \}_{1 \leq i \leq n}$ consisting of distinct elements of $P$ , is free. There exists an $\alpha \in I$ and elements $x_\alpha^{(i)} \in P_\alpha$ such that $y^{(i)} = f_\alpha(x_\alpha^{(i)})$ for $1 \leq i \leq n$ (Set Theory, III, § 7, no. 5, Lemma 1); if $\sum_i \lambda^{(i)} y^{(i)} = 0$ , it may be assumed that $\lambda^{(i)} = \phi_\alpha(\lambda_\alpha^{(i)})$ for $1 \leq i \leq n$ and hence $f_\alpha \left( \sum_i \lambda_\beta^{(i)} x_\beta^{(i)} \right) = 0$ ; this implies $\sum_i \lambda_\beta^{(i)} x_\beta^{(i)} = 0$ for some $\beta \geq \alpha$ , where $\lambda_\beta^{(i)} = \phi_{\beta \alpha}(\lambda_\alpha^{(i)})$ , $x_\beta^{(i)} = f_{\beta \alpha}(x_\alpha^{(i)})$ and the $x_\beta^{(i)}$ belong to $P_\beta$ and are distinct since $y^{(i)} = f_\beta(x_\beta^{(i)})$ ; then $\lambda_\beta^{(i)} = 0$ for $1 \leq i \leq n$ , whence

\lambda^{(i)} = \phi_\beta(\lambda_\beta^{(i)}) = 0

for

1 \leq i \leq n

Suppose now that all the rings $A_\alpha$ are equal to the same ring $A$ and the $\phi_{\beta \alpha}$ to $1_A$ ; then, for every direct system $(E_\alpha, f_{\beta \alpha})$ of $A$ -modules, $E = \lim \overrightarrow{E_\alpha}$ is an $A$ -module. Let $F$ be an $A$ -module and for all $\alpha$ let $u_\alpha : E_\alpha \to F$ be an $A$ -linear mapping such that $(u_\alpha)$ is a direct system of mappings; then $u = \lim \overrightarrow{u_\alpha}$ is an $A$ -linear mapping of $E$ into $F$ . Conversely, for every $A$ -linear mapping $v : \lim \overrightarrow{E_\alpha} \to F$ , the family of $v_\alpha = v \circ f_\alpha$ is a direct system of $A$ -linear mappings such that $v = \lim \overrightarrow{v_\alpha}$ . On the other hand we note that for $\alpha \leq \beta$ the mapping

\operatorname{Hom}(f_{\beta \alpha}, 1_F) = \bar{f}_{\alpha \beta} : \operatorname{Hom}_A(E_\beta, F) \to \operatorname{Hom}_A(E_\alpha, F)

is a

\mathbf{Z}

-module homomorphism such that

(\operatorname{Hom}_A(E_\alpha, F), \bar{f}_{\alpha \beta})

is an inverse system of

\mathbf{Z}

*-modules*; as

\bar{f}_{\alpha \beta}(v_\beta) = v_\beta \circ f_{\beta \alpha}

, the above remarks can be expressed as follows:

Proposition 6

For every direct system* $(E_\alpha, f_{\beta \alpha})$ of $A$ *-modules and every* $A$ *-module* $F$ , the canonical mapping $u \mapsto (u \circ f_\alpha)$ is a $\mathbf{Z}$ *-module isomorphism*

d_F : \operatorname{Hom}_A(\lim \overrightarrow{E_\alpha}, F) \to \lim \leftarrow \operatorname{Hom}_A(E_\alpha, F).

Corollary 1

For every* $A$ *-module homomorphism* $v : F \to F'$ , the

\bar{v}_\alpha = \operatorname{Hom}(1_{E_\alpha}, v) : \operatorname{Hom}(E_\alpha, F) \to \operatorname{Hom}(E_\alpha, F')

form an inverse system of $\mathbf{Z}$ -linear mappings and the diagram

\begin{array}{ccc} \operatorname{Hom}(\lim E_\alpha, F) & \xrightarrow{d_F} & \lim \operatorname{Hom}(E_\alpha, F) \\ \downarrow \operatorname{Hom}(1_E, v) & & \downarrow \lim \overline{v}_\alpha \\ \operatorname{Hom}(\lim E_\alpha, F') & \xrightarrow{d_{F'}} & \lim \operatorname{Hom}(E_\alpha, F') \end{array}

is commutative.

For all $u \in \operatorname{Hom}(\lim E_\alpha, F)$ , $d_{F'}(v \circ u) = (v \circ u \circ f_\alpha)$ by definition and the commutativity of diagram (6) then follows immediately from the definitions.

Corollary 2

If* $(E_\alpha, f_{\beta \alpha})$ is a direct system of left $A$ *-modules and* $E = \lim \longrightarrow E_\alpha$ , $(E_\alpha^*, t f_{\beta \alpha})$ is an inverse system of right $A$ *-modules and* $\lim \leftarrow E_\alpha^*$ is canonically isomorphic to $E^*$ .

Remark

Let $E$ be an $A$ *-module and* $(M_\alpha)_{\alpha \in I}$ an increasing family of submodules of $E$ such that $E$ is the union of the $M_\alpha$ ; if $j_{\beta \alpha}: M_\alpha \to M_\beta$ (for $\alpha \leq \beta$ ) and $j_\alpha: M_\alpha \to E$ are the canonical injections, it is immediate that $j = \lim \longrightarrow j_\alpha$ is an isomorphism of $\lim \longrightarrow M_\alpha$ onto $E$ (Set Theory, III, § 7, no. 6, Remark 1). In particular, every $A$ *-module is the direct limit of the right directed family of its finitely generated submodules.

3. Tensor product of direct limits

Let $(A_\alpha, \rho_{\beta \alpha})$ be a direct system of rings and $(E_\alpha, f_{\beta \alpha})$ (resp. $(F_\alpha, g_{\beta \alpha})$ ) be a direct system of right (resp. left) $A_\alpha$ *-modules. For $\alpha \leq \beta$ , there is a $\mathbf{Z}$ *-module homomorphism*

f_{\beta \alpha} \otimes g_{\beta \alpha}: E_\alpha \otimes_{A_\alpha} F_\alpha \to (E_\beta)_{[A_\alpha]} \otimes_{A_\alpha} (F_\beta)_{[A_\alpha]}

and on the other hand there is a canonical $\mathbf{Z}$ *-module homomorphism*

(E_\beta)_{[A_\alpha]} \otimes_{A_\alpha} (F_\beta)_{[A_\alpha]} \to E_\beta \otimes_{A_\beta} F_\beta

corresponding to the ring homomorphism $\rho_{\beta \alpha}$ ( $§ 3$ , no. 3, Proposition 2); whence by composition we obtain a $\mathbf{Z}$ *-module homomorphism*

h_{\beta \alpha}: E_\alpha \otimes_{A_\alpha} F_\alpha \to E_\beta \otimes_{A_\beta} F_\beta

which maps the tensor product $x_\alpha \otimes y_\alpha$ to $f_{\beta \alpha}(x_\alpha) \otimes g_{\beta \alpha}(y_\alpha)$ . Clearly

(E_\alpha \otimes_{A_\alpha} F_\alpha, h_{\beta \alpha})

is a direct system of $\mathbf{Z}$ *-modules. Let* $A = \lim \longrightarrow A_\alpha$ , $E = \lim \longrightarrow E_\alpha$ , $F = \lim \longrightarrow F_\alpha$ and

let $\rho_\alpha : A_\alpha \to A, f_\alpha : E_\alpha \to E, g_\alpha : F_\alpha \to F$ be the canonical mappings. As above, a $\mathbf{Z}$ -linear mapping $\pi_\alpha : E_\alpha \otimes_{A_\alpha} F_\alpha \to E \otimes_A F$ is defined, which maps the tensor product $x_\alpha \otimes y_\alpha$ to $f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha)$ , and it is immediate that these mappings form a direct system. Thus we obtain a $\mathbf{Z}$ -linear mapping

\pi = \lim \pi_\alpha : \lim (E_\alpha \otimes_{A_\alpha} F_\alpha) \to E \otimes_A F.

Proposition 7

The $\mathbf{Z}$ -linear mapping (7) is bijective.

We set $P = \lim (E_\alpha \otimes_{A_\alpha} F_\alpha)$ and, for all $\alpha \in I$ , let $h_\alpha : E_\alpha \otimes_{A_\alpha} F_\alpha \to P$ be the canonical mapping. On the other hand, for all $\alpha \in I$ , let

t_\alpha : E_\alpha \times F_\alpha \to E_\alpha \otimes_{A_\alpha} F_\alpha

be the canonical

\mathbf{Z}

-bilinear mapping; for

\alpha \leq \beta

t_\beta(f_{\beta \alpha}(x_\alpha), g_{\beta \alpha}(y_\alpha)) = f_{\beta \alpha}(x_\alpha) \otimes g_{\beta \alpha}(y_\alpha) = h_{\beta \alpha}(t_\alpha(x_\alpha, y_\alpha))

and hence

(t_\alpha)

is a direct system of mappings. Canonically identifying

\lim (E_\alpha \times F_\alpha)

with

E \times F

(Set Theory, III, § 7, no. 7, Proposition 10), we derive a mapping

t = \lim t_\alpha : E \times F \to P

such that

t(f_\alpha(x_\alpha), g_\alpha(y_\alpha)) = h_\alpha(t_\alpha(x_\alpha, y_\alpha)) = h_\alpha(x_\alpha \otimes y_\alpha).

Taking account of Set Theory, III, § 7, no. 5, Lemma 1, it is immediately seen that

t

\mathbf{Z}

-bilinear; moreover, for

x \in E, y \in F, \lambda \in A

, there exists

\alpha \in I

such that

x = f_\alpha(x_\alpha), y = g_\alpha(y_\alpha), \lambda = \rho_\alpha(\lambda_\alpha)

with

\lambda_\alpha \in A_\alpha, x_\alpha \in E_\alpha, y_\alpha \in F_\alpha

(Set Theory, III, § 3, no. 7, Lemma 1); whence

t(x \lambda, y) = h_\alpha((x_\alpha \lambda_\alpha) \otimes y_\alpha) = h_\alpha(x_\alpha \otimes (\lambda_\alpha y_\alpha)) = t(x, \lambda y).

Hence there exists one and only one

\mathbf{Z}

-linear mapping

\pi' : E \otimes_A F \to P

such that

\pi'(x \otimes y) = t(x, y)

(§ 3, no. 1, Proposition 1). Moreover, by definition

\pi'(\pi(h_\alpha(x_\alpha \otimes y_\alpha))) = \pi'(f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha)) = h_\alpha(x_\alpha \otimes y_\alpha)

\pi(\pi'(f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha))) = \pi(h_\alpha(x_\alpha \otimes y_\alpha)) = f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha)

and as the elements of the form

f_\alpha(x_\alpha) \otimes g_\alpha(y_\alpha)

(resp.

h_\alpha(x_\alpha \otimes y_\alpha)

) generate the

\mathbf{Z}

-module

E \otimes_A F

(resp.

P

\pi' \circ \pi

and

\pi \circ \pi'

are the identity mappings.

Loosely speaking, Proposition 7 may be expressed by saying that tensor products commute with direct limits and usually the two sides of (7) are identified by means of the isomorphism $\pi$ .

Corollary 1

Let* $(E'_\alpha, f'_{\beta \alpha})$ (resp. $(F'_\alpha, g'_{\alpha \beta})$ ) be another direct system of right (resp. left) $A_\alpha$ -*modules; for all* $\alpha \in I$ , let $u_\alpha : E_\alpha \to E'_\alpha$ (resp. $v_\alpha : F_\alpha \to F'_\alpha$ ) *be an

$A_\alpha$ -linear mapping such that $(u_\alpha)$ (resp. $(v_\alpha)$ ) is a direct system. Then $(u_\alpha \oplus v_\alpha)$ is a direct system of $\mathbf{Z}$ -linear mappings and the diagram

\begin{array}{ccc} \lim(E_\alpha \otimes_{A_\alpha} F_\alpha) & \longrightarrow & (\lim E_\alpha) \otimes_A (\lim F_\alpha) \\ \downarrow & & \downarrow \\ \lim(u_\alpha \otimes v_\alpha) & & (\lim u_\alpha) \otimes (\lim v_\alpha) \\ \lim(E'_\alpha \otimes_{A_\alpha} F'_\alpha) & \longrightarrow & (\lim E'_\alpha) \otimes_A (\lim F'_\alpha) \end{array}

is commutative.

The verification is immediate.

Let $(A'_\alpha, \rho'_{\beta\alpha})$ be another direct system of rings and suppose that each $E_\alpha$ is an $(A'_\alpha, A_\alpha)$ -bimodule, the $f_{\beta\alpha}$ being $(A'_\alpha, A_\alpha)$ -linear for $\alpha \leq \beta$ . Then if we write $A' = \lim A'_\alpha$ , the isomorphism (7) is linear with respect to the left $A'$ -module structures on the two sides by virtue of Corollary 1. This can be immediately generalized to arbitrary multimodules.

In particular, if the $A_\alpha$ are commutative, $A = \lim A_\alpha$ is commutative and isomorphism (7) is an A-module isomorphism.

Corollary 2

Let* $(E_\alpha, f_{\beta\alpha})$ be a direct system of right $A_\alpha$ *-modules and let* $E'_\alpha = E_\alpha \otimes_{A_\alpha} A$ be the $A$ *-module obtained by extending the ring of scalars to* $A = \lim A_\alpha$ by means of the canonical homomorphism $\rho_\alpha : A_\alpha \to A$ . Then $(E'_\alpha, f_{\beta\alpha} \otimes 1_A)$ is a direct system of right $A$ *-modules, whose direct limit is canonically isomorphic to $\lim E_\alpha$ .

It suffices to apply Proposition 7 with $F_\alpha$ the ring $A$ considered as an $(A_\alpha, A)$ -bimodule by means of $\rho_\alpha$ .

Corollary 3

Let* $A$ be a ring, $(E_\alpha, f_{\beta\alpha})$ a direct system of right $A$ *-modules and* $F$ a left $A$ *-module. Then the $\mathbf{Z}$ *-modules* $\lim (E_\alpha \otimes_A F)$ and $(\lim E_\alpha) \otimes_A F$ are canonically isomorphic.

It suffices to take $A_\alpha = A$ and $F_\alpha = F$ for all $\alpha \in I$ in Proposition 7.

In particular, if $\rho : A \to B$ is a ring homomorphism, $\lim \rho^*(E_\alpha)$ and $\rho^*(\lim E_\alpha)$ are canonically isomorphic.

Corollary 4

Let* $M$ be a right $A$ *-module, $N$ a left $A$ *-module, $(x_i)_{1 \leq i \leq n}$ a family of elements of $M$ , $(y_i)_{1 \leq i \leq n}$ a family of elements of $N$ , such that $\sum_i (x_i \otimes y_i) = 0$ in $M \otimes_A N$ . Then there exists a finitely generated submodule $M_1$ (resp. $N_1$ ) of $M$ (resp. $N$ ) containing the $x_i$ (resp. the $y_i$ ) and such that $\sum_i (x_i \otimes y_i) = 0$ in $M_1 \otimes_A N_1$ .

$M$ (resp. $N$ ) is canonically identified with the direct limit of the right directed family of its finitely generated submodules containing the $x_i$ (resp. the $y_i$ ) and it suffices to apply *Set Theory, III, § 7, no. 5, Lemma 1.

Exercises

See the exercises for § 6.