Corollary

For every projective A-module E, the B-module $\rho^*(E)$ is projective. If further $\rho$ is injective, the canonical mapping of E into $\rho^*(E)$ is injective.

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