Corollary

Under the hypotheses of Proposition 3, for an $A$*-module* $M$ to be finitely generated (resp. finitely presented), it is necessary and sufficient that, for every index $i$, the $A_{f_i}$*-module* $M_{f_i}$ be finitely generated (resp. finitely presented).

The condition is obviously necessary (§ 2, no. 4). Conversely, if all the $M_{f_i}$ are finitely generated (resp. finitely presented), $M' = \prod_{i \in I} M_{f_i}$ is a finitely generated (resp. finitely presented) $B$-module, for we can obviously suppose that for each $i$ there is an exact sequence $AT_i \to A^n_{f_i} \to M_{f_i} \to 0$, where $m$ and $n$ are independent of $i$. Now, $M' = M \otimes_A B$. The corollary then follows from Proposition 3 and Chapter I, § 3, no. 6, Proposition 11.

Note that the condition on the $f_i$ means that the open sets $X_{f_i}$ form a covering of $\mathrm{Spec}(A)$ (§ 4, no. 3, Corollary 3 to Proposition 11).