Lemma 10.95.4 (05A7)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.95: Descending properties of modules
Lemma 10.95.4 (cite)

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Lemma 10.95.4. Let $R \to S$ be a ring map, let $M$ be an $R$ -module, and let $Q$ be a countably generated $S$ -submodule of $M \otimes _ R S$ . Then there exists a countably generated $R$ -submodule $P$ of $M$ such that $\mathop{\mathrm{Im}}(P \otimes _ R S \to M \otimes _ R S)$ contains $Q$ .

Proof. Let $y_1, y_2, \ldots$ be generators for $Q$ and write $y_ j = \sum _ k x_{jk} \otimes s_{jk}$ for some $x_{jk} \in M$ and $s_{jk} \in S$ . Then take $P$ be the submodule of $M$ generated by the $x_{jk}$ . $\square$

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