Example 67.9.6. Strange example of a universally closed morphism. Let $\mathbf{Q} \subset k$ be a field of characteristic zero. Let $X = \mathbf{A}^1_ k/\mathbf{Z}$ as in Spaces, Example 65.14.8. We claim the structure morphism $p : X \to \mathop{\mathrm{Spec}}(k)$ is universally closed. Namely, if $Z/k$ is a scheme, and $T \subset |X \times _ k Z|$ is closed, then $T$ corresponds to a $\mathbf{Z}$-invariant closed subset of $T' \subset |\mathbf{A}^1 \times Z|$. It is easy to see that this implies that $T'$ is the inverse image of a subset $T''$ of $Z$. By Morphisms, Lemma 29.25.12 we have that $T'' \subset Z$ is closed. Of course $T''$ is the image of $T$. Hence $p$ is universally closed by Lemma 67.9.5.

## 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Your email address will not be published.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)