The Stacks project

Lemma 5.23.7. 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. 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 5.23.5), 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). 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 (1)

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).


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