Lemma 77.4.8 (0CWM)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.4: Flat finite type modules
Lemma 77.4.8 (cite)

previous theorem

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Lemma 77.4.8. Let $S$ be a local scheme with closed point $s$. Let $f : X \to S$ be a morphism from an algebraic space $X$ to $S$ which is 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 77.4.7. $\square$

[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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