Lemma 33.7.16. Let $k \subset K$ be a finite extension of fields and let $X$ be a scheme over $k$. Denote by $p : X_ K \to X$ the projection morphism. For every connected component $T$ of $X_ K$ the image $p(T)$ is a connected component of $X$.

Proof. The image $p(T)$ is contained in some connected component $X'$ of $X$. Consider $X'$ as a closed subscheme of $X$ in any way. Then $T$ is also a connected component of $X'_ K = p^{-1}(X')$ and we may therefore assume that $X$ is connected. The morphism $p$ is open (Morphisms, Lemma 29.23.4), closed (Morphisms, Lemma 29.44.7) and the fibers of $p$ are finite sets (Morphisms, Lemma 29.44.10). Thus we may apply Topology, Lemma 5.7.7 to conclude. $\square$

## 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Your email address will not be published.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.

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 07VM. Beware of the difference between the letter 'O' and the digit '0'.