Theorem 38.26.1. Let $f : X \to S$ be locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $x \in X$ with image $s \in S$. The following are equivalent

$\mathcal{F}$ is flat at $x$ over $S$, and

for every $x' \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$ which specializes to $x$ we have that $\mathcal{F}$ is flat at $x'$ over $S$.

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