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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