Lemma 29.20.17. Let $X \to Y$ be a morphism of schemes over a base scheme $S$. Let $x \in X$. If $X \to S$ is quasi-finite at $x$, then $X \to Y$ is quasi-finite at $x$. If $X$ is locally quasi-finite over $S$, then $X \to Y$ is locally quasi-finite.

Proof. Via Lemma 29.20.11 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is quasi-finite, then $B \to C$ is quasi-finite. This follows from Algebra, Lemma 10.122.6 with $R = A$, $S = S' = C$ and $R' = B$. $\square$

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