Lemma 38.13.7 (05IL)—The Stacks project

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Part 2: Schemes
Chapter 38: More on Flatness
Section 38.13: Flat finite type modules, Part II
Lemma 38.13.7 (cite)

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Lemma 38.13.7. Let $R \to S$ be a ring map of finite presentation. Let $M$ be a finite $S$ -module. Assume $\text{WeakAss}_ S(S)$ 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 a finitely presented $S_ g$ -module flat over $R$ .

Proof. Follows immediately from Theorem 38.13.6. $\square$

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