The Stacks project

Lemma 64.22.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. Let $(U, \overline{u})$ be an étale neighbourhood of $\overline{x}$ where $U$ is a scheme. Then we have

\[ \mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, \overline{u}} = \mathcal{O}_{U, u}^{sh} \]

where the left hand side is the stalk of the structure sheaf of $X$, and the right hand side is the strict henselization of the local ring of $U$ at the point $u$ at which $\overline{u}$ is centered.

Proof. We know that the structure sheaf $\mathcal{O}_ U$ on $U_{\acute{e}tale}$ is the restriction of the structure sheaf of $X$. Hence the first equality follows from Lemma 64.19.9 part (4). The second equality is explained in √Čtale Cohomology, Lemma 58.33.1. $\square$


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