The Stacks project

Lemma 5.23.8. Let $X$ be a spectral space. The following are equivalent:

  1. $X$ is profinite,

  2. $X$ is Hausdorff,

  3. $X$ is totally disconnected,

  4. every quasi-compact open is closed,

  5. there are no nontrivial specializations between points,

  6. every point of $X$ is closed,

  7. every point of $X$ is the generic point of an irreducible component of $X$,

  8. the constructible topology equals the given topology on $X$, and

  9. add more here.

Proof. Lemma 5.22.2 shows the implication (1) $\Rightarrow $ (3). Irreducible components are closed, so if $X$ is totally disconnected, then every point is closed. So (3) implies (6). The equivalence of (6) and (5) is immediate, and (6) $\Leftrightarrow $ (7) holds because $X$ is sober. Assume (5). Then all constructible subsets of $X$ are closed (Lemma 5.23.6), in particular all quasi-compact opens are closed. So (5) implies (4). Since $X$ is sober, for any two points there is a quasi-compact open containing exactly one of them, hence (4) implies (2). Parts (4) and (8) are equivalent by the definition of the constructible topology. It remains to prove (2) implies (1). Suppose $X$ is Hausdorff. Every quasi-compact open is also closed (Lemma 5.12.4). This implies $X$ is totally disconnected. Hence it is profinite, by Lemma 5.22.2. $\square$

Comments (2)

Comment #4225 by Laurent Moret-Bailly on

Suggestion for (8): the constructible topology is the original one. (Obviously, this is another way to state (4) but it could be worth mentioning).

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Your email address will not be published.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 0905. Beware of the difference between the letter 'O' and the digit '0'.