Lemma 29.28.7. Let $f : X \to S$ be a morphism of schemes. Let $x \leadsto x'$ be a nontrivial specialization of points in $X$ lying over the same point $s \in S$. Assume $f$ is locally of finite type. Then

$\dim _ x(X_ s) \leq \dim _{x'}(X_ s)$,

$\dim (\mathcal{O}_{X_ s, x}) < \dim (\mathcal{O}_{X_ s, x'})$, and

$\text{trdeg}_{\kappa (s)}(\kappa (x)) > \text{trdeg}_{\kappa (s)}(\kappa (x'))$.

## Comments (1)

Comment #734 by Keenan Kidwell on