The Stacks project

10.94 Ascending properties of modules

All of the properties of a module in Theorem 10.93.3 ascend along arbitrary ring maps:

Lemma 10.94.1. Let $R \to S$ be a ring map. Let $M$ be an $R$-module. Then:

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

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

  3. 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.

  4. 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 10.88.6 and the fact that tensoring commutes with taking colimits. $\square$

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