Lemma 27.2.3. Let $X$ be a quasi-separated scheme. The intersection of any two quasi-compact opens of $X$ is a quasi-compact open of $X$. Every quasi-compact open of $X$ is retrocompact in $X$.

Proof. If $U$ and $V$ are quasi-compact open then $U \cap V = \Delta ^{-1}(U \times V)$, where $\Delta : X \to X \times X$ is the diagonal. As $X$ is quasi-separated we see that $\Delta$ is quasi-compact. Hence we see that $U \cap V$ is quasi-compact as $U \times V$ is quasi-compact (details omitted; use Schemes, Lemma 25.17.4 to see $U \times V$ is a finite union of affines). The other assertions follow from the first and Topology, Lemma 5.27.1. $\square$

## 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Your email address will not be published.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 054D. Beware of the difference between the letter 'O' and the digit '0'.