The Stacks project

Lemma 68.11.8. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. Let $(U, u) \to (X, x)$ be an elementary étale neighbourhood. Then

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

In words: the henselian local ring of $X$ at $x$ is equal to the henselization $\mathcal{O}_{U, u}^ h$ of the local ring $\mathcal{O}_{U, u}$ of $U$ at $u$.

Proof. Since the category of elementary étale neighbourhood of $(X, x)$ is cofiltered (Lemma 68.11.6) we see that the category of elementary étale neighbourhoods of $(U, u)$ is initial in the category of elementary étale neighbourhood of $(X, x)$. Then the equality follows from More on Morphisms, Lemma 37.35.5 and Categories, Lemma 4.17.2 (initial is turned into cofinal because the colimit definining henselian local rings is over the opposite of the category of elementary étale neighbourhoods). $\square$

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