Lemma 33.7.19. Let $k$ be a field, with separable algebraic closure $\overline{k}$. Let $X$ be a scheme over $k$. Assume
$X$ is quasi-compact, and
the connected components of $X_{\overline{k}}$ are open.
Then
$\pi _0(X_{\overline{k}})$ is finite, and
the action of $\text{Gal}(\overline{k}/k)$ on $\pi _0(X_{\overline{k}})$ is continuous.
Moreover, assumptions (1) and (2) are satisfied when $X$ is of finite type over $k$.
Proof. Since the connected components are open, cover $X_{\overline{k}}$ (Topology, Lemma 5.7.3) and $X_{\overline{k}}$ is quasi-compact, we conclude that there are only finitely many of them. Thus (a) holds. By Lemma 33.7.8 these connected components are each defined over a finite subextension of $\overline{k}/k$ and we get (b). If $X$ is of finite type over $k$, then $X_{\overline{k}}$ is of finite type over $\overline{k}$ (Morphisms, Lemma 29.15.4). Hence $X_{\overline{k}}$ is a Noetherian scheme (Morphisms, Lemma 29.15.6). Thus $X_{\overline{k}}$ has finitely many irreducible components (Properties, Lemma 28.5.7) and a fortiori finitely many connected components (which are therefore open). $\square$
Comments (0)