$B$ can be considered as a $(B, A)$-bimodule by means of $\rho$; then there is a canonical $\mathbf{Z}$-module isomorphism

(1)

as has been seen in $§ 4$, no. 1, Proposition 1. But the left $A$-module $\operatorname{Hom}_B(B_s, F)$ is canonically identified with $\rho^*(F)$: for, by definition ($§ 1$, no. 14), there corresponds to an element $y \in F$ the homomorphism $\theta(y) : B_s \to F$ such that $(\theta(y))(1) = y$; for all $\lambda \in A$, there thus corresponds to $\rho(\lambda)y \in F$ the homomorphism $\mu \mapsto \mu \rho(\lambda)y$ of $B_s$ into $F$, which is just $\lambda \theta(y)$ for the left $A$-module structure on $\operatorname{Hom}_B(B_s, F)$ ($§ 1$, no. 14). Using this identification, we obtain therefore a canonical $\mathbf{Z}$-module isomorphism, the inverse of (1)

(2)

and it follows immediately from the definitions that if $\delta(f) = \bar{f}$ then $\bar{f}(1 \otimes x) = f(x)$ for all $x \in E$. In particular, the mapping $\phi_E : x \mapsto 1 \otimes x$ is just (3)

Proposition 1 is therefore proved. The mapping $\phi_E : E \to \rho_*(\rho^*(E))$ is called canonical.

where $\phi_E$ and $\phi_{E'}$ are the canonical mappings.

It suffices to apply Proposition 1 to the A-homomorphism $\phi_{E'} \circ u : E \to E'_{(B)}$.

The mapping $v$ defined in the above corollary is denoted by $\rho^*(u)$ or $u_{(B)}$.

If $E''$ is a third left A-module and $v : E' \to E''$ an A-linear mapping, it is immediate that

Extending the ring of operators of a module is a transitive operation; to be precise:

mapping $1 \otimes (1 \otimes x)$ to $1 \otimes x$ for all $x \in E$ and this homomorphism is bijective.

The underlying $\mathbf{Z}$-modules of $\sigma^*(\rho^*(E))$ and $(\sigma \circ \rho)^*(E)$ are respectively $C \otimes_B (B \otimes_A E)$ and $C \otimes_A E$. There exists a canonical $\mathbf{Z}$-isomorphism $C \otimes_B (B \otimes_A E) \to (C \otimes_B B) \otimes_A E$ ($§ 3$, no. 8, Proposition 8), which is also a C-isomorphism for the left C-module structures on both sides. Moreover, the

C-module $C \otimes_B B$ is canonically identified with the C-module $C_s$ under the isomorphism which maps $\gamma \otimes \beta$ to $\gamma \sigma(\beta)$ (§ 3, no. 4, Proposition 4) and this isomorphism is also an isomorphism for the right A-module structure on $C \otimes_B B$ defined by $\rho$ and the right A-module structure on C defined by $\sigma \circ \rho$. Thus a canonical isomorphism

is obtained and, composing it with the isomorphism

defined earlier, the desired canonical isomorphism is obtained.

If $\phi, \phi'$ and $\phi''$ denote the canonical mappings $E \to \rho^*(E), \rho^*(E \to) \sigma^*(\rho^*(E))$ and $E \to (\sigma \circ \rho)^*(E)$, $\phi' \circ \phi$ is identified with $\phi''$ under the canonical isomorphism of Proposition 2.

mapping $(1 \otimes x) \otimes (1 \otimes x')$ to $1 \otimes (x \otimes x')$ for $x \in E, x' \in E'$, and this homomorphism is bijective.

The left hand side of (5) may be written $(B \otimes_A E) \otimes_B (B \otimes_A E')$ and is identified with $(E \otimes_A B) \otimes_B (B \otimes_A E')$ since A and B are commutative; the latter product is identified successively with $E \otimes_A (B \otimes_B B) \otimes_A E'$, $E \otimes_A (B \otimes_A E')$, $E \otimes_A (E' \otimes_A B)$ and finally $(E \otimes_A E') \otimes_A B$, using the associativity of the tensor product (§ 3, no. 8, Proposition 8), Proposition 4, § 3, no. 4, and the commutativity of A and B. The desired isomorphism is the composition of the successive canonical isomorphisms.

Clearly if S is a generating system of E, the image of S under the canonical mapping $E \to E_{(B)}$ is a generating system of $E_{(B)}$; in particular, if E is a finitely generated A-module, $E_{(B)}$ is a finitely generated B-module.

The first assertion follows immediately from § 3, no. 7, Corollary 1 to Proposition 7. Also, for every family $(\xi_\lambda)_{\lambda \in L}$ of elements of A of finite support,

and the relation $\phi \left( \sum_{\lambda \in L} \xi_\lambda a_\lambda \right) = 0$ is therefore equivalent to $\rho(\xi_\lambda) = 0$ for all $\lambda \in L$, whence the second assertion.

By hypothesis there exists a free A-module M containing E and in which E admits a supplement F. It follows immediately from § 3, no. 7, Proposition 7 that $M_{(B)}$ is identified with the direct sum of $E_{(B)}$ and $F_{(B)}$ and if $\phi$ and $\psi$ are the canonical mappings $E \to E_{(B)}$ and $F \to F_{(B)}$, the canonical mapping $M \to M_{(B)}$ is just $x + y \mapsto \phi(x) + \psi(y)$. The corollary follows immediately from Proposition 4 applied to the A-module M.

When E is a right A-module, we write similarly $\rho^*(E) = E \otimes_A \rho_*(B_s)$, B being considered this time as an (A, B)-bimodule and the right B-module structure on $\rho^*(E)$ being such that $(x \otimes \beta)\beta' = x \otimes (\beta\beta')$ for $\beta \in B' \ \beta' \in B$ and $x \in E$. We leave to the reader the task of stating for right modules the results corresponding to those of this no. and the following.

Remark (4). Consider the left A-module $\rho_*(B_s)$ defined by $\rho$ and for every left A-module E, consider the $\mathbf{Z}$-module

As $\rho_*(B_s)$ has a right B-module structure, a left B-module structure is derived on $\tilde{\rho}(E)$ (§ 1, no. 14) such that, if $u \in \tilde{\rho}(E)$ and $b' \in B$, $b'u$ is the homomorphism $b \mapsto u(bb')$ of $\rho_*(B_s)$ into E. We further define an A-linear mapping, called canonical associating with every homomorphism $u \in \tilde{\rho}(E)$ the element $u(1)$ in E. As B can be considered as an (A, B)-bimodule by means of $\rho$, for every left B-module F, there is a canonical $\mathbf{Z}$-module isomorphism (§ 1, no. 1, Proposition 1). As the left A-module $\rho^*(B_s) \otimes_B F$ is canonically identified with $\rho_*(F)$ by virtue of § 3, no. 4, Proposition 4, we obtain a canonical $\mathbf{Z}$-module isomorphism, the inverse of the above which associates with every B-linear mapping g of F into $\tilde{\rho}(E)$ the composite mapping $\eta \circ g$, considered as an A-linear mapping of $\rho_*(F)$ into E. In particular, under the hypotheses of Proposition 2, if F is replaced by $\sigma_*(C_s)$, we obtain a canonical C-isomorphism

are respectively equal to the identity mappings of $\rho^*(E)$ and $\rho_*(F)$.

We give the proof, for example, for (12); for all $x \in E$, the mapping $\rho^*(\phi_E)$ maps $1 \otimes x$ to the element $1 \otimes (1 \otimes x)$ and the mapping $\psi_{\rho^*(E)}$ maps $1 \otimes (1 \otimes x)$ to the element $1 \otimes x$; the conclusion follows from the fact that the elements of the form $1 \otimes x$ generate the B-module $\rho^*(E)$; the proof is even simpler for (13).

This is a consequence of Proposition 5 and § 1, no. 9, Corollary 2 to Proposition 15.

mapping $y \otimes x$ to $y \otimes (1 \otimes x)$ for all $x \in E$ and all $y \in F$ and this homomorphism is bijective.

By definition the right hand side of (14) is $F \otimes_B (B \otimes_A E)$, where B is considered as a (B, A)-bimodule, and there is a canonical $\mathbf{Z}$-isomorphism $(F \otimes_B B) \otimes_A E \to F \otimes_B (B \otimes_A E)$ defined in § 3, no. 8, Proposition 8; on the other hand, the canonical isomorphism $F \to F \otimes_B B$ of § 3, no. 4, Proposition 4 is an isomorphism for the right A-module structures on the two sides, defined by $\rho$. Whence the desired isomorphism.

When A and B are commutative, the isomorphism (14) is an A-module isomorphism

(i) If B is a projective (resp. finitely generated projective) A-module, the homomorphism (17) is injective* (resp. bijective).

(ii) If E is a finitely generated projective A-module, the homomorphism* (17) is bijective.

As (16) is bijective, the proposition follows from § 4, no. 2. Proposition 2, applied to the canonical homomorphism (15).

This follows from the above and § 4, no. 2, Proposition 2.

Suppose in particular that E is a finitely generated free A-module and let $(e_i)_{1 \leq i \leq n}$ be a basis of E and $(e_i^*)$ the dual basis; then the canonical isomorphism (19) maps the basis $(e_i^* \otimes 1)$ of $(E^*)_{(B)}$ to the dual basis of the basis $(1 \otimes e_i)$ of $E_{(B)}$.

We know (§ 2, no. 6, Proposition 12) that there exists a family $(a_\lambda)_{\lambda \in L}$ of elements of P and a family $(a_\lambda^*)_{\lambda \in L}$ of elements of the dual P* such that, for all $x \in P$, the family $\langle x, a_\lambda^* \rangle$ is of finite support and $x = \sum_\lambda \langle x, a_\lambda^* \rangle a_\lambda$. Since $P_{(B)}$ is finitely generated, there exists a finite family $(y_i)_{i \in I}$ of elements of P such that $P_{(B)}$ is generated by the elements $1 \otimes y_i$. For each index i, the family $\langle y_i, a_\lambda^* \rangle$ has finite support. Hence there exists a finite subset H of L such that $\langle y_i, a_\lambda^* \rangle = 0$ for $i \in I$ and $\lambda \notin H$. Since

it follows that $1 d_B \otimes a_\lambda^* = 0$ for $\lambda \notin H$. Hence, for all $x \in P$, for $\lambda \notin H$. This shows that the A-module P is generated by the $a_\lambda$ such that $\lambda \in H$.