The Stacks project

Lemma 38.6.6. Let $R \to S$ be a finite type ring map. Let $M$ be a finite $S$-module. Let $\mathfrak q$ be a prime ideal of $S$. There exists an elementary étale localization $R' \to S', \mathfrak q', \mathfrak p'$ of the ring map $R \to S$ at $\mathfrak q$ such that there exists a complete dévissage of $(M \otimes _ S S')/S'/R'$ at $\mathfrak q'$.

Proof. This is a reformulation of Proposition 38.5.7 via Remark 38.6.5 $\square$

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