## 10.93 Ascending properties of modules

All of the properties of a module in Theorem 10.92.3 ascend along arbitrary ring maps:

Lemma 10.93.1. Let $R \to S$ be a ring map. Let $M$ be an $R$-module. Then:

If $M$ is flat, then the $S$-module $M \otimes _ R S$ is flat.

If $M$ is Mittag-Leffler, then the $S$-module $M \otimes _ R S$ is Mittag-Leffler.

If $M$ is a direct sum of countably generated $R$-modules, then the $S$-module $M \otimes _ R S$ is a direct sum of countably generated $S$-modules.

If $M$ is projective, then the $S$-module $M \otimes _ R S$ is projective.

**Proof.**
All are obvious except (2). For this, use formulation (3) of being Mittag-Leffler from Proposition 10.87.6 and the fact that tensoring commutes with taking colimits.
$\square$

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