Lemma 38.13.12 (05IR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.13: Flat finite type modules, Part II
Lemma 38.13.12: Algebra version of Remark 38.13.11 (cite)

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Lemma 38.13.12. Let $R \to S$ be a ring map of finite type. Let $M$ be a finite $S$ -module. Assume $\text{WeakAss}_ R(R)$ is finite. Then

$U = \{ \mathfrak q \subset S \mid M_{\mathfrak q}\text{ flat over }R\}$

is open in $\mathop{\mathrm{Spec}}(S)$ and for every $g \in S$ such that $D(g) \subset U$ the localization $M_ g$ is flat over $R$ and an $S_ g$ -module finitely presented relative to $R$ (see More on Algebra, Definition 15.80.2).

Proof. This is Lemma 38.13.11 translated into algebra. $\square$

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