The Stacks project

Lemma 21.31.1. The category $\textit{LC}$ has fibre products and a final object and hence has arbitrary finite limits. Given morphisms $X \to Z$ and $Y \to Z$ in $\textit{LC}$ with $X$ and $Y$ quasi-compact, then $X \times _ Z Y$ is quasi-compact.

Proof. The final object is the singleton space. Given morphisms $X \to Z$ and $Y \to Z$ of $\textit{LC}$ the fibre product $X \times _ Z Y$ is a subspace of $X \times Y$. Hence $X \times _ Z Y$ is Hausdorff as $X \times Y$ is Hausdorff by Topology, Section 5.3.

If $X$ and $Y$ are quasi-compact, then $X \times Y$ is quasi-compact by Topology, Theorem 5.14.4. Since $X \times _ Z Y$ is a closed subset of $X \times Y$ (Topology, Lemma 5.3.4) we find that $X \times _ Z Y$ is quasi-compact by Topology, Lemma 5.12.3.

Finally, returning to the general case, if $x \in X$ and $y \in Y$ we can pick quasi-compact neighbourhoods $x \in E \subset X$ and $y \in F \subset Y$ and we find that $E \times _ Z F$ is a quasi-compact neighbourhood of $(x, y)$ by the result above. Thus $X \times _ Z Y$ is an object of $\textit{LC}$ by Topology, Lemma 5.13.2. $\square$

Comments (0)

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Your email address will not be published.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 09WZ. Beware of the difference between the letter 'O' and the digit '0'.