Lemma 10.142.5. Let $R \to S$ be a ring map. Let $\mathfrak q \subset S$ be a prime lying over $\mathfrak p$ in $R$. If $S/R$ is étale at $\mathfrak q$ then

we have $\mathfrak p S_{\mathfrak q} = \mathfrak qS_{\mathfrak q}$ is the maximal ideal of the local ring $S_{\mathfrak q}$, and

the field extension $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ is finite separable.

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