Lemma 37.18.4. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module flat over $S$. Assume $S$ is irreducible with generic point $\eta $. If $\dim (\text{Supp}(\mathcal{F}_\eta )) \leq r$ then for all $s \in S$ we have $\dim (\text{Supp}(\mathcal{F}_ s)) \leq r$.
Proof. Let $x \in \text{Supp}(\mathcal{F}_ s)$ be a generic point of an irreducible component of $\text{Supp}(\mathcal{F}_ s)$. By Algebra, Lemma 10.41.12 we can find a specialization $y \leadsto x$ in $\text{Supp}(\mathcal{F})$ with $f(y) = \eta $. Of course we may assume $y$ is a generic point of an irreducible component of $\text{Supp}(\mathcal{F}_\eta )$. We conclude from Lemma 37.18.3 that the dimension of $\overline{\{ x\} }$ is at most $r$. $\square$
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