Lemma 10.143.6. An étale ring map is quasi-finite.
Proof. Let $R \to S$ be an étale ring map. By definition $R \to S$ is of finite type. For any prime $\mathfrak p \subset R$ the fibre ring $S \otimes _ R \kappa (\mathfrak p)$ is étale over $\kappa (\mathfrak p)$ and hence a finite products of fields finite separable over $\kappa (\mathfrak p)$, in particular finite over $\kappa (\mathfrak p)$. Thus $R \to S$ is quasi-finite by Lemma 10.122.4. $\square$
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