Lemma 66.25.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The following are equivalent
$X$ is regular, and
every étale local ring $\mathcal{O}_{X, \overline{x}}$ is regular.
66.25 Regular algebraic spaces
We have already defined regular algebraic spaces in Section 66.7.
Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. By assumption $U$ is locally Noetherian. Moreover, every étale local ring $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of a local ring on $U$ and conversely, see Lemma 66.22.1. Thus by More on Algebra, Lemma 15.45.10 we see that (2) is equivalent to every local ring of $U$ being regular, i.e., $U$ being a regular scheme (see Properties, Lemma 28.9.2). This equivalent to (1) by Definition 66.7.2. $\square$
We can use Descent, Lemma 35.21.4 to define what it means for an algebraic space $X$ to be regular at a point $x$.
Definition 66.25.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. We say $X$ is regular at $x$ if $\mathcal{O}_{U, u}$ is a regular local ring for any (equivalently some) pair $(a : U \to X, u)$ consisting of an étale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$.
See Definition 66.7.5, Lemma 66.7.4, and Descent, Lemma 35.21.4.
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 $x$.
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 66.22.1. Thus we conclude by More on Algebra, Lemma 15.45.10. $\square$
Lemma 66.25.4. A regular algebraic space is normal.
Proof. This follows from the definitions and the case of schemes See Properties, Lemma 28.9.4. $\square$
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