Lemma 38.18.1. Let $f : X \to S$ be a morphism of schemes of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Assume that $\mathcal{F}$ is flat over $S$ at all points of $X_ s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$ with $f(x') = s'$ such that $s' \leadsto s$ is a specialization in $S$. If $x'$ specializes to a point of $X_ s$, then $x' \leadsto x$ with $x \in \text{Ass}_{X_ s}(\mathcal{F}_ s)$.

Proof. Say $x' \leadsto t$ with $t \in X_ s$. Then we can find specializations $x' \leadsto x \leadsto t$ with $x$ corresponding to a generic point of an irreducible component of $\overline{\{ x'\} } \cap f^{-1}(\{ s\} )$. By assumption $\mathcal{F}$ is flat over $S$ at $x$. By More on Morphisms, Lemma 37.18.3 we see that $x \in \text{Ass}_{X/S}(\mathcal{F})$ as desired. $\square$

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