Lemma 10.147.6. Let $R \to S$ be a finite type ring map. Let $\mathfrak q$ be a prime of $S$. If $R \to S$ is unramified at $\mathfrak q$ then $R \to S$ is quasi-finite at $\mathfrak q$. In particular, an unramified ring map is quasi-finite.
Proof. An unramified ring map is of finite type. Thus it is clear that the second statement follows from the first. To see the first statement apply the characterization of Lemma 10.121.2 part (2) using Lemma 10.147.5. $\square$
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