The Stacks project

Lemma 10.14.2. 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 of 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 10.12.3 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. $\square$

Comments (2)

Comment #5398 by Laurent Moret-Bailly on

To make sense of (1), we need an -module structure on , which presupposes a ring structure on . But I could not find the definition of a tensor products of algebras.

Comment #5630 by on

We do say in Definition 10.14.1 that is a module over , so it does get mentioned. The Stacks project is not like Bourbaki and it assumes you know some things before you start reading, but it isn't entirely clear what the things are you already have to know. I guess we could add this to the section on basic notions in this chapter. Anybody who wants to code this up, please go ahead (but please make it just a couple of lines).

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View Lemma 10.14.2 as pdf