Lemma 10.143.5 (00U4)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.143: Étale ring maps
Lemma 10.143.5 (cite)

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

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 $\kappa (\mathfrak q)/\kappa (\mathfrak p)$ 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 $\kappa (\mathfrak p)$ . $\square$

The Stacks project

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