The Stacks project

Lemma 38.14.4. Let $R$ be a henselian local ring with maximal ideal $\mathfrak m$. Let $R \to S$ be a ring map. Let $N$ be an $S$-module. Assume $N$ is countably generated and Mittag-Leffler as an $R$-module. Then for any $R$-module $M$ and for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes _ R M$ we have $\mathfrak q + \mathfrak m S \not= S$.

Proof. This lemma reduces to Lemma 38.14.3 by Algebra, Lemma 10.153.13. $\square$

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