The Stacks project

Proof. Proof of (1). Take a surjective, $S$-linear map $S^{\oplus n} \to M \to 0$. By Lemma 10.12.3 and 10.12.10 the result after tensoring with $R^\prime$ is a surjection ${S^\prime }^{\oplus n} \to M^\prime \rightarrow 0$, so $M^\prime$ is a finitely generated $S^\prime$-module. Proof of (2). Take a presentation $S^{\oplus m} \to S^{\oplus n} \to M \to 0$. By Lemma 10.12.3 and 10.12.10 the result after tensoring with $R^\prime$ gives a finite presentation ${S^\prime }^{\oplus m} \to {S^\prime }^{\oplus n} \to M^\prime \to 0$, of the $S^\prime$-module $M^\prime$. Proof of (3). This follows by the remark preceding the lemma as we can take $I$ to be finite by assumption. Proof of (4). This follows by the remark preceding the lemma as we can take $I$ and $J$ to be finite by assumption. $\square$