The Stacks project

Lemma 5.23.15. Let $X$ be a topological space and let $c : X \to X'$ be the universal map from $X$ to a sober topological space, see Lemma 5.8.16.

  1. If $X$ is quasi-compact, so is $X'$.

  2. If $X$ is quasi-compact, has a basis of quasi-compact opens, and the intersection of two quasi-compact opens is quasi-compact, then $X'$ is spectral.

  3. If $X$ is Noetherian, then $X'$ is a Noetherian spectral space.

Proof. Let $U \subset X$ be open and let $U' \subset X'$ be the corresponding open, i.e., the open such that $c^{-1}(U') = U$. Then $U$ is quasi-compact if and only if $U'$ is quasi-compact, as pulling back by $c$ is a bijection between the opens of $X$ and $X'$ which commutes with unions. This in particular proves (1).

Proof of (2). It follows from the above that $X'$ has a basis of quasi-compact opens. Since $c^{-1}$ also commutes with intersections of pairs of opens, we see that the intersection of two quasi-compact opens $X'$ is quasi-compact. Finally, $X'$ is quasi-compact by (1) and sober by construction. Hence $X'$ is spectral.

Proof of (3). It is immediate that $X'$ is Noetherian as this is defined in terms of the acc for open subsets which holds for $X$. We have already seen in (2) that $X'$ is spectral. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 5.23: Spectral spaces

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Your email address will not be published.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 0A2T. Beware of the difference between the letter 'O' and the digit '0'.