The Stacks project

Definition 66.22.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over the point $x \in |X|$.

  1. The étale local ring of $X$ at $\overline{x}$ is the stalk of the structure sheaf $\mathcal{O}_ X$ on $X_{\acute{e}tale}$ at $\overline{x}$. Notation: $\mathcal{O}_{X, \overline{x}}$.

  2. The strict henselization of $X$ at $\overline{x}$ is the scheme $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}})$.

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