Proposition 20

(i) For a finitely generated A-module M to be pseudo-zero, it is necessary and sufficient that $M \otimes_A B$ be a pseudo-zero B-module. (ii) If M is a finitely generated torsion A-module, $M \otimes_A B$ is a finitely generated B-module and: (22)

(iii) If M is a finitely generated A-module, $M \otimes_A B$ is a finitely generated B-module and: (23) (24)

(i) Let $\mathfrak{p}$ be a prime ideal of $B$, $\mathfrak{p} = \mathfrak{P} \cap A$; then $(M \otimes_A B)_{\mathfrak{p}} = M \otimes_A B_{\mathfrak{p}}$ (Chapter 11, § 2, no. 7, Proposition 18) and on the other hand

the relation $M_p = 0$ is therefore equivalent to $(M \otimes_A B)_{\mathfrak{p}} = 0$ (Chapter II, § 4, no. 4, Lemma 4). It suffices to apply this remark to the ideal $\mathfrak{P} = (0)$ and the ideals $\mathfrak{p} \in P(B)$ to prove (i), taking account of no. 4, Definition 2.

To prove (ii), we shall use the following lemma: