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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0B8E. Beware of the difference between the letter 'O' and the digit '0'.