The Stacks project

Lemma 78.10.1. Let $k$ be a field with algebraic closure $\overline{k}$. Let $G$ be a group algebraic space over $k$ which is separated1. Then $G_{\overline{k}}$ is a scheme.

Proof. By Spaces over Fields, Lemma 71.10.2 it suffices to show that $G_ K$ is a scheme for some field extension $K/k$. Denote $G_ K' \subset G_ K$ the schematic locus of $G_ K$ as in Properties of Spaces, Lemma 65.13.1. By Properties of Spaces, Proposition 65.13.3 we see that $G_ K' \subset G_ K$ is dense open, in particular not empty. Choose a scheme $U$ and a surjective ├ętale morphism $U \to G$. By Varieties, Lemma 33.14.2 if $K$ is an algebraically closed field of large enough transcendence degree, then $U_ K$ is a Jacobson scheme and every closed point of $U_ K$ is $K$-rational. Hence $G_ K'$ has a $K$-rational point and it suffices to show that every $K$-rational point of $G_ K$ is in $G_ K'$. If $g \in G_ K(K)$ is a $K$-rational point and $g' \in G_ K'(K)$ a $K$-rational point in the schematic locus, then we see that $g$ is in the image of $G_ K'$ under the automorphism

\[ G_ K \longrightarrow G_ K,\quad h \longmapsto g(g')^{-1}h \]

of $G_ K$. Since automorphisms of $G_ K$ as an algebraic space preserve $G_ K'$, we conclude that $g \in G_ K'$ as desired. $\square$

[1] It is enough to assume $G$ is decent, e.g., locally separated or quasi-separated by Lemma 78.9.4.

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