The Stacks project

Remark 38.24.2. Here is a scheme theoretic reformulation of Theorem 38.24.1. Let $(X, x) \to (S, s)$ be a morphism of pointed schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $S$ henselian local with closed point $s$. 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

  1. $\mathcal{F}_ T$ is flat over $T$ at all points of the fibre $X_ t$ which map to $x \in X_ s$, and

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

Moreover, if $X \to S$ is of finite presentation at $x$ and $\mathcal{F}_ x$ of finite presentation over $\mathcal{O}_{X, x}$, then $Z \to S$ is of finite presentation.

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