The Stacks project

Lemma 64.25.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent

  1. $X$ is regular at $x$, and

  2. the étale local ring $\mathcal{O}_{X, \overline{x}}$ is regular for any (equivalently some) geometric point $\overline{x}$ lying over $x$.

Proof. Let $U$ be a scheme, $u \in U$ a point, and let $a : U \to X$ be an étale morphism mapping $u$ to $x$. For any geometric point $\overline{x}$ of $X$ lying over $x$, the étale local ring $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of a local ring on $U$ at $u$, see Lemma 64.22.1. Thus we conclude by More on Algebra, Lemma 15.45.10. $\square$

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