The Stacks project

Lemma 10.95.2. Let $R \to S$ be a faithfully flat ring map. Let $M$ be an $R$-module. If the $S$-module $M \otimes _ R S$ is countably generated, then $M$ is countably generated.

Proof. Say $M \otimes _ R S$ is generated by the elements $y_ i$, $i = 1, 2, 3, \ldots $. Write $y_ i = \sum _{j = 1, \ldots , n_ i} x_{ij} \otimes s_{ij}$ for some $n_ i \geq 0$, $x_{ij} \in M$ and $s_{ij} \in S$. Denote $M' \subset M$ the submodule generated by the countable collection of elements $x_{ij}$. Then $M' \otimes _ R S \to M \otimes _ R S$ is surjective as the image contains the generators $y_ i$. Since $S$ is faithfully flat over $R$ we conclude that $M' = M$ as desired. $\square$

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