The Stacks project

Lemma 66.25.1. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The following are equivalent

  1. $X$ is regular, and

  2. every étale local ring $\mathcal{O}_{X, \overline{x}}$ is regular.

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. $\square$

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