The Stacks project

Lemma 10.143.5. 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

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

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

Proof. First we may replace $S$ by $S_ g$ for some $g \in S$, $g \not\in \mathfrak q$ and assume that $R \to S$ is étale. Then the lemma follows from Lemma 10.143.4 by unwinding the fact that $S \otimes _ R \kappa (\mathfrak p)$ is étale over $\kappa (\mathfrak p)$. $\square$

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