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

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

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

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

Proof. Part (1) follows from the fact that any open of $X_ s$ containing $x'$ also contains $x$. Part (2) follows since $\mathcal{O}_{X_ s, x}$ is a localization of $\mathcal{O}_{X_ s, x'}$ at a prime ideal, hence any chain of prime ideals in $\mathcal{O}_{X_ s, x}$ is part of a strictly longer chain of primes in $\mathcal{O}_{X_ s, x'}$. The last inequality follows from Algebra, Lemma 10.116.2. $\square$

Comment #734 by Keenan Kidwell on

The second $\mathscr{O}_{X_s,x}$ of the second line of the proof should be $\mathscr{O}_{X_s,x^\prime}$. Also, "at a prime ideal" instead of "in a prime ideal?"

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