Lemma 66.34.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume $f$ is locally of finite type. Then we have

where the notation is as in Definition 66.33.1.

Lemma 66.34.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume $f$ is locally of finite type. Then we have

\[ \begin{matrix} \text{relative dimension of }f\text{ at }x
\\ =
\\ \text{dimension of local ring of the fibre of }f\text{ at }x
\\ +
\\ \text{transcendence degree of }x/f(x)
\end{matrix} \]

where the notation is as in Definition 66.33.1.

**Proof.**
This follows immediately from Morphisms, Lemma 29.28.1 applied to $h : U \to V$ and $u \in U$ as in Lemma 66.22.5.
$\square$

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