The Stacks project

Lemma 70.4.2. Let $S$ be a scheme. Let $X$ be an integral algebraic space over $S$. Let $\eta \in |X|$ be the generic point of $X$. There are canonical identifications

\[ R(X) = \mathcal{O}_{X, \eta }^ h = \kappa (\eta ) \]

where $R(X)$ is the ring of rational functions defined in Morphisms of Spaces, Definition 65.47.3, $\kappa (\eta )$ is the residue field defined in Decent Spaces, Definition 66.11.2, and $\mathcal{O}_{X, \eta }^ h$ is the henselian local ring defined in Decent Spaces, Definition 66.11.5. In particular, these rings are fields.

Proof. Since $X$ is a scheme in an open neighbourhood of $\eta $ (see discussion above), this follows immediately from the corresponding result for schemes, see Morphisms, Lemma 29.48.5. We also use: the henselianization of a field is itself and that our definitions of these objects for algebraic spaces are compatible with those for schemes. Details omitted. $\square$


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