Lemma 66.33.3. Let $S$ be a scheme. Let $X \to Y \to Z$ be morphisms of algebraic spaces over $S$. Let $x \in |X|$ and let $y \in |Y|$, $z \in |Z|$ be the images. Assume $X \to Y$ is locally quasi-finite and $Y \to Z$ locally of finite type. Then the transcendence degree of $x/z$ is equal to the transcendence degree of $y/z$.

**Proof.**
We can choose commutative diagrams

where $U, V, W$ are schemes and the vertical arrows are étale. By definition the morphism $U \to V$ is locally quasi-finite which implies that $\kappa (v) \subset \kappa (u)$ is finite, see Morphisms, Lemma 29.20.5. Hence the result is clear. $\square$

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