Lemma 67.33.3. 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 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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