Lemma 38.26.2 (05UB)—The Stacks project

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Lemma 38.26.2. Let $S$ be a local scheme with closed point $s$ . Let $f : X \to S$ be locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$ -module. Assume that

every point of $\text{Ass}_{X/S}(\mathcal{F})$ specializes to a point of the closed fibre $X_ s$ ¹,
$\mathcal{F}$ is flat over $S$ at every point of $X_ s$ .

Then $\mathcal{F}$ is flat over $S$ .

Proof. This is immediate from the fact that it suffices to check for flatness at points of the relative assassin of $\mathcal{F}$ over $S$ by Theorem 38.26.1. $\square$

[1] For example this holds if

$f$ is finite type and

$\mathcal{F}$ is pure along

$X_ s$ , or if

$f$ is proper.

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