Lemma 76.4.8. 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 $S$.

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