The Stacks project

Lemma 29.20.18. Let $f : X \to Y$ and $g : Y \to S$ be morphisms of schemes. If $f$ is surjective, $g \circ f$ locally quasi-finite, and $g$ locally of finite type, then $g : Y \to S$ is locally quasi-finite.

Proof. Let $x \in X$ with images $y \in Y$ and $s \in S$. Since $g \circ f$ is locally quasi-finite by Lemma 29.20.5 the extension $\kappa (x)/\kappa (s)$ is finite. Hence $\kappa (y)/\kappa (s)$ is finite. Hence $y$ is a closed point of $Y_ s$ by Lemma 29.20.2. Since $f$ is surjective, we see that every point of $Y$ is closed in its fibre over $S$. Thus by Lemma 29.20.6 we conclude that $g$ is quasi-finite at every point. $\square$

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