Lemma 10.158.8 (0321)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.158: Formal smoothness of fields
Lemma 10.158.8 (cite)

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Formally smooth equals separable for field extensions.

Lemma 10.158.8. Let $k$ be a field.

If the characteristic of $k$ is zero, then any extension field of $k$ is formally smooth over $k$ .
If the characteristic of $k$ is $p > 0$ , then $K/k$ is formally smooth if and only if it is a separable field extension.

Proof. Combine Lemmas 10.158.5 and 10.158.7. $\square$

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Comment #1034 by Matthew Emerton on September 10, 2014 at 01:03

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