Lemma 66.25.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent
$X$ is regular at $x$, and
the étale local ring $\mathcal{O}_{X, \overline{x}}$ is regular for any (equivalently some) geometric point $\overline{x}$ lying over $x$.
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