The Stacks project

Lemma 66.33.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite type, $Y$ is Jacobson (Properties of Spaces, Remark 65.7.3), and $x \in |X|$ is a finite type point of $X$, then the transcendence degree of $x/f(x)$ is $0$.

Proof. Choose a scheme $V$ and a surjective ├ętale morphism $V \to Y$. Choose a scheme $U$ and a surjective ├ętale morphism $U \to X \times _ Y V$. By Lemma 66.25.5 we can find a finite type point $u \in U$ mapping to $x$. After shrinking $U$ we may assume $u \in U$ is closed (Morphisms, Lemma 29.16.4). Let $v \in V$ be the image of $u$. By Morphisms, Lemma 29.16.8 the extension $\kappa (u)/\kappa (v)$ is finite. This finishes the proof. $\square$

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