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.

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

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

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

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

## Comments (2)

Comment #5398 by Laurent Moret-Bailly on

Comment #5630 by Johan on