Lemma 33.25.8 (0B8X)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.25: Schemes smooth over fields
Lemma 33.25.8 (cite)

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Lemma 33.25.8. Let $k$ be a perfect field. Let $X$ be a locally algebraic reduced $k$ -scheme, for example a variety over $k$ . Then we have

$\{ x \in X \mid X \to \mathop{\mathrm{Spec}}(k)\text{ is smooth at }x\} = \{ x \in X \mid \mathcal{O}_{X, x}\text{ is regular}\}$

and this is a dense open subscheme of $X$ .

Proof. The equality of the two sets follows immediately from Algebra, Lemma 10.140.5 and the definitions (see Algebra, Definition 10.45.1 for the definition of a perfect field). The set is open because the set of points where a morphism of schemes is smooth is open, see Morphisms, Definition 29.34.1. Finally, we give two arguments to see that it is dense: (1) The generic points of $X$ are in the set as the local rings at generic points are fields (Algebra, Lemma 10.25.1) hence regular. (2) We use that $X$ is geometrically reduced by Lemma 33.6.3 and hence Lemma 33.25.7 applies. $\square$

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