Lemma 10.158.5 (031Y)—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 10.158.5 (cite)

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Lemma 10.158.5. Let $K/k$ be an extension of fields. If $K$ is formally smooth over $k$ , then $K$ is a separable extension of $k$ .

Proof. Assume $K$ is formally smooth over $k$ . By Lemma 10.138.9 we see that $K \otimes _ k \Omega _{k/\mathbf{Z}} \to \Omega _{K/\mathbf{Z}}$ is injective. Hence $K$ is separable over $k$ by Lemma 10.158.4. $\square$

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