The Stacks project

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.