Lemma 77.10.1. Let $S$ be the spectrum of a henselian local ring with closed point $s$. Let $X \to S$ be a morphism of algebraic spaces which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $E \subset |X_ s|$ be a subset. There exists a closed subscheme $Z \subset S$ with the following property: for any morphism of pointed schemes $(T, t) \to (S, s)$ the following are equivalent

$\mathcal{F}_ T$ is flat over $T$ at all points of $|X_ t|$ which map to a point of $E \subset |X_ s|$, and

$\mathop{\mathrm{Spec}}(\mathcal{O}_{T, t}) \to S$ factors through $Z$.

Moreover, if $X \to S$ is locally of finite presentation, $\mathcal{F}$ is of finite presentation, and $E \subset |X_ s|$ is closed and quasi-compact, then $Z \to S$ is of finite presentation.

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