Proposition 11

Let $B$ be a ring and $A$ a subring of $B$ such that $B$ is a faithfully flat right $A$-*module. For a left* $A$-*module* $F$ to be definitely generated (resp. finitely presented), it is necessary and sufficient that the $B$-*module* $B \otimes_A F$ be finitely generated (resp. finitely presented).

(1) Without any hypothesis on $B$, clearly, if $F$ is a finitely generated left $A$-module, $B \otimes_A F$ is a finitely generated left $B$-module. Conversely, if $B \otimes_A F$ is a finitely generated $B$-module, it is generated by a finite number of elements of the form $1 \otimes x_i$ with $x_i \in F$; if $M$ is a sub-A-module of $F$ generated by the $x_i$ and $j$ the canonical injection $M \to F$, $1, \otimes j : B \otimes_A M \to B \otimes_A F$ is a surjective homomorphism, hence $j$ is surjective (no. 1, Proposition 2), which proves that $F$ is finitely generated.

(2) If $F$ admits a finite presentation, so does $B \otimes_A F$ without any hypothesis on B (§ 2, no. 8). It remains to prove that, if $B \otimes_A F$ admits a finite presentation, so does F. We already know from (1) that F is finitely generated, hence there exists a surjective homomorphism $u : L \to F$, where L is a finitely generated free A-module. Let R be the kernel of u, so that $B \otimes_A R$ is identified with the kernel of the surjective homomorphism $l_B \otimes u : B \otimes_A L \to B \otimes_A F$ (§ 2, no. 3, Remark 2). As $B \otimes_A F$ admits a finite presentation by hypothesis, we conclude (§ 2, no. 8, Lemma 9) that $B \otimes_A R$ is finitely generated; then it follows from (1) that R is a finitely generated A-module and consequently F admits a finite presentation.