The Stacks project

Lemma 5.24.7. Let $W$ be a subset of a spectral space $X$. The following are equivalent:

  1. $W$ is an intersection of constructible sets and closed under generalizations,

  2. $W$ is quasi-compact and closed under generalizations,

  3. there exists a quasi-compact subset $E \subset X$ such that $W$ is the set of points specializing to $E$,

  4. $W$ is an intersection of quasi-compact open subsets,

  5. there exists a nonempty set $I$ and quasi-compact opens $U_ i \subset X$, $i \in I$ such that $W = \bigcap U_ i$ and for all $i, j \in I$ there exists a $k \in I$ with $U_ k \subset U_ i \cap U_ j$.

In this case we have (a) $W$ is a spectral space, (b) $W = \mathop{\mathrm{lim}}\nolimits U_ i$ as topological spaces, and (c) for any open $U$ containing $W$ there exists an $i$ with $U_ i \subset U$.

Proof. Let $W \subset X$ satisfy (1). Then $W$ is closed in the constructible topology, hence quasi-compact in the constructible topology (by Lemmas 5.23.2 and 5.12.3), hence quasi-compact in the topology of $X$ (because opens in $X$ are open in the constructible topology). Thus (2) holds.

It is clear that (2) implies (3) by taking $E = W$.

Let $X$ be a spectral space and let $E \subset W$ be as in (3). Since every point of $W$ specializes to a point of $E$ we see that an open of $W$ which contains $E$ is equal to $W$. Hence since $E$ is quasi-compact, so is $W$. If $x \in X$, $x \not\in W$, then $Z = \overline{\{ x\} }$ is disjoint from $W$. Since $W$ is quasi-compact we can find a quasi-compact open $U$ with $W \subset U$ and $U \cap Z = \emptyset $. We conclude that (4) holds.

If $W = \bigcap _{j \in J} U_ j$ then setting $I$ equal to the set of finite subsets of $J$ and $U_ i = U_{j_1} \cap \ldots \cap U_{j_ r}$ for $i = \{ j_1, \ldots , j_ r\} $ shows that (4) implies (5). It is immediate that (5) implies (1).

Let $I$ and $U_ i$ be as in (5). Since $W = \bigcap U_ i$ we have $W = \mathop{\mathrm{lim}}\nolimits U_ i$ by the universal property of limits. Then $W$ is a spectral space by Lemma 5.24.5. Let $U \subset X$ be an open neighbourhood of $W$. Then $E_ i = U_ i \cap (X \setminus U)$ is a family of constructible subsets of the spectral space $Z = X \setminus U$ with empty intersection. Using that the spectral topology on $Z$ is quasi-compact (Lemma 5.23.2) we conclude from Lemma 5.12.6 that $E_ i = \emptyset $ for some $i$. $\square$


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Your email address will not be published.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 0A31. Beware of the difference between the letter 'O' and the digit '0'.