Lemma 37.20.10. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. Suppose $x' \leadsto x$ is a specialization of points of $X$ with image $s' \leadsto s$ in $S$. If $x$ is a generic point of an irreducible component of $X_ s$ then $\dim _{x'}(X_{s'}) = \dim _ x(X_ s)$.

Proof. The point $x$ is contained in $U_ d$ for some $d$, where $U_ d$ as in Lemma 37.20.9. $\square$

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