The Stacks project

Lemma 68.11.9. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over $x \in |X|$. The ├ętale local ring $\mathcal{O}_{X, \overline{x}}$ of $X$ at $\overline{x}$ (Properties of Spaces, Definition 66.22.2) is the strict henselization of the henselian local ring $\mathcal{O}_{X, x}^ h$ of $X$ at $x$.

Proof. Follows from Lemma 68.11.8, Properties of Spaces, Lemma 66.22.1 and the fact that $(R^ h)^{sh} = R^{sh}$ for a local ring $(R, \mathfrak m, \kappa )$ and a given separable algebraic closure $\kappa ^{sep}$ of $\kappa $. This equality follows from Algebra, Lemma 10.154.7. $\square$

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