Lemma 39.7.15. Let $G$ be a group scheme over a field. There exists an open and closed subscheme $G' \subset G$ which is a countable union of affines.

**Proof.**
Let $e \in U(k)$ be a quasi-compact open neighbourhood of the identity element. By replacing $U$ by $U \cap i(U)$ we may assume that $U$ is invariant under the inverse map. As $G$ is separated this is still a quasi-compact set. Set

where $m_ n : G \times _ k \ldots \times _ k G \to G$ is the $n$-slot multiplication map $(g_1, \ldots , g_ n) \mapsto m(m(\ldots (m(g_1, g_2), g_3), \ldots ), g_ n)$. Each of these maps are open (see Lemma 39.7.1) hence $G'$ is an open subgroup scheme. By Lemma 39.7.7 it is also a closed subgroup scheme. $\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: