The Stacks project

Lemma 68.11.10. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$. The residue field of the henselian local ring of $X$ at $x$ (Definition 68.11.7) is the residue field of $X$ at $x$ (Definition 68.11.2).

Proof. Choose an elementary ├ętale neighbourhood $(U, u) \to (X, x)$. Then $\kappa (u) = \kappa (x)$ and $\mathcal{O}_{X, x}^ h = \mathcal{O}_{U, u}^ h$ (Lemma 68.11.8). The residue field of $\mathcal{O}_{U, u}^ h$ is $\kappa (u)$ by Algebra, Lemma 10.155.1 (the output of this lemma is the construction/definition of the henselization of a local ring, see Algebra, Definition 10.155.3). $\square$

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