The Stacks project

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. $\square$

Comments (2)

Comment #2448 by Andy on

What is supposed to mean?

Comment #2490 by on

When is a point of a scheme, then we think of as a scheme too, namely we think of as the spectrum of its residue field. So what is meant is that the base change to reduces the lemma to the case where the base scheme is the spectrum of a field.

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View Lemma 29.28.1 as pdf