The Stacks project

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

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ X \ar[r] & Y \ar[r] & Z } \quad \quad \xymatrix{ u \ar[d] \ar[r] & v \ar[d] \ar[r] & w \ar[d] \\ x \ar[r] & y \ar[r] & z } \]

where $U, V, W$ are schemes and the vertical arrows are ├ętale. 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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