Lemma 10.94.1. 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.

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