Proposition 12

Let V be a left B-module. The B-module V is projective (resp. generating, faithful, $*$injective, finitely presented ) if and only if the

A-module $P\otimes_BV$ is.

a) Suppose that V is projective. There exists a set I such that V is isomorphic to a direct factor submodule of $B^{(I)}_s$ . The A-module $P\otimes_BV$ is then isomorphic to a direct factor submodule of $P^{(I)}$; since P is a projective A-module, the same holds for $P\otimes_BV$.

b) Suppose that the B-module V is generating. Let M be an A-module. There exists a B-module W such that M is isomorphic to $P\otimes_BW$. By Theorem 1 of VIII, p. 80, there exist a set I and a surjection $\varphi : V^{(I)}\rightarrow W$. By the corollary, the mapping $1_P\otimes \varphi$ from $P\otimes (V^{(I})$ to $P\otimes_AW$ is surjective, which gives a surjection $(P\otimes V)^{(I)}\rightarrow M$. By Theorem 1 of VIII, p. 80, $P\otimes V$ is a generating A-module.

c) The B-module V is faithful if and only if its annihilator is reduced to 0. Assertion c) therefore follows from Example 3 of VIII, p. 105.

$*$d) Suppose that V is injective. By the remark of VIII, p. 106, the A-module $P\otimes_BV$ is isomorphic to Hom$_B(Q,V)$, where Q is a $(B,A)_k$-bimodule inverse to A. Since the A-module Q is projective, hence flat (X, §1, n$^o3$, p. 9, exemple 1), the A-module Hom$_B(Q,V)$ is injective by X, §1, n$^o8$, p. 18, proposition 11.

e) Suppose that V admits a finite presentation $L_1\rightarrow L_0\rightarrow V\rightarrow 0$ (X, §1, n$^o4$, p. 10). By taking the tensor product with P, we deduce an exact sequence of A-modules $N'_1-\overset{u}{\rightarrow}N'_0\rightarrow P\otimes_BV\rightarrow 0$, where $N'_1$ and $N'_0$ are projective and finitely generated (Proposition 11 and a)). Let $N''_0$ be a finitely generated A-module such that the module $N_0= N'_0\oplus N''_0$ is free and finitely generated, and let $u': N'_1\oplus N''_0\rightarrow N_0$ be the homomorphism $(u,1_{N''_0})$; then $P\otimes_BV$ can be identified with the cokernel of $u'$. Let $N_1$ be a finitely generated free A-module and $p: N_1\rightarrow N'_1\oplus N''_0$ a surjective homomorphism; the sequence $N_1\longrightarrow^{u'\circ p}N_0\rightarrow P\otimes_BV\rightarrow 0$ is a finite presentation of the A-module $P\otimes_BV.*$

f) Suppose that the A-module $P\otimes_BV$ is projective (resp. generating, faithful, $*$injective, finitely presented ). By applying the above (interchanging

the roles of A and B and of P and Q), we see that the B-module $Q\otimes_AP\otimes_BV$ also has this property. The same is therefore true for the B-module V, which is isomorphic to it.