Lemma 10.158.10 (037X)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.158: Formal smoothness of fields
Lemma 10.158.10 (cite)

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Lemma 10.158.10. Let $K/k$ be a finitely generated field extension. Then $K$ is separable over $k$ if and only if $K$ is the localization of a smooth $k$ -algebra.

Proof. Choose a finite type $k$ -algebra $R$ which is a domain whose fraction field is $K$ . Lemma 10.140.9 says that $k \to R$ is smooth at $(0)$ if and only if $K/k$ is separable. This proves the lemma. $\square$

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