Lemma 29.28.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ and set $s = f(x)$. Assume $f$ is locally of finite type. Then

\[ \dim _ x(X_ s) = \dim (\mathcal{O}_{X_ s, x}) + \text{trdeg}_{\kappa (s)}(\kappa (x)). \]

Lemma 29.28.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ and set $s = f(x)$. Assume $f$ is locally of finite type. Then

\[ \dim _ x(X_ s) = \dim (\mathcal{O}_{X_ s, x}) + \text{trdeg}_{\kappa (s)}(\kappa (x)). \]

**Proof.**
This immediately reduces to the case $S = s$, and $X$ affine. In this case the result follows from Algebra, Lemma 10.116.3.
$\square$

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