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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