
Lemma 10.13.2. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. Let $R \to R'$ be a ring map and let $S' = S \otimes _ R R'$ and $M' = M \otimes _ R R'$ be the base changes.

1. If $M$ is a finite $S$-module, then the base change $M'$ is a finite $S'$-module.

2. If $M$ is an $S$-module finite presentation, then the base change $M'$ is an $S'$-module of finite presentation.

3. If $R \to S$ is of finite type, then the base change $R' \to S'$ is of finite type.

4. If $R \to S$ is of finite presentation, then the base change $R' \to S'$ is of finite presentation.

Proof. Proof of (1). Take a surjective, $S$-linear map $S^{\oplus n} \to M \to 0$. By Lemma 10.11.3 and 10.11.10 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 10.11.3 and 10.11.10 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. $\square$

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