Lemma 15.18.1. Let $R \to S$ be a ring map. Let $I \subset R$ be an ideal. Let $M$ be an $S$-module. Let $R \to R'$ be a ring map and $IR' \subset I' \subset R'$ an ideal. If (15.18.0.1) holds for $(R \to S, I, M)$, then (15.18.0.1) holds for $(R' \to S \otimes _ R R', I', M \otimes _ R R')$.

Proof. Assume (15.18.0.1) holds for $(R \to S, I \subset R, M)$. Let $I'(S \otimes _ R R') \subset \mathfrak q'$ be a prime of $S \otimes _ R R'$. Let $\mathfrak q \subset S$ be the corresponding prime of $S$. Then $IS \subset \mathfrak q$. Note that $(M \otimes _ R R')_{\mathfrak q'}$ is a localization of the base change $M_{\mathfrak q} \otimes _ R R'$. Hence $(M \otimes _ R R')_{\mathfrak q'}$ is flat over $R'$ as a localization of a flat module, see Algebra, Lemmas 10.39.7 and 10.39.18. $\square$

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