The Stacks project

Lemma 5.24.4. Let $\mathcal{I}$ be a cofiltered category. Let $i \mapsto X_ i$ be a diagram of spectral spaces such that for $a : j \to i$ in $\mathcal{I}$ the corresponding map $f_ a : X_ j \to X_ i$ is spectral. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with projections $p_ i : X \to X_ i$. Let $E \subset X$ be a constructible subset. Then there exists an $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a constructible subset $E_ i \subset X_ i$ such that $p_ i^{-1}(E_ i) = E$. If $E$ is open, resp. closed, we may choose $E_ i$ open, resp. closed.

Proof. Assume $E$ is a quasi-compact open of $X$. By Lemma 5.14.2 we can write $E = p_ i^{-1}(U_ i)$ for some $i$ and some open $U_ i \subset X_ i$. Write $U_ i = \bigcup U_{i, \alpha }$ as a union of quasi-compact opens. As $E$ is quasi-compact we can find $\alpha _1, \ldots , \alpha _ n$ such that $E = p_ i^{-1}(U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n})$. Hence $E_ i = U_{i, \alpha _1} \cup \ldots \cup U_{i, \alpha _ n}$ works.

Assume $E$ is a constructible closed subset. Then $E^ c$ is quasi-compact open. So $E^ c = p_ i^{-1}(F_ i)$ for some $i$ and quasi-compact open $F_ i \subset X_ i$ by the result of the previous paragraph. Then $E = p_ i^{-1}(F_ i^ c)$ as desired.

If $E$ is general we can write $E = \bigcup _{l = 1, \ldots , n} U_ l \cap Z_ l$ with $U_ l$ constructible open and $Z_ l$ constructible closed. By the result of the previous paragraphs we may write $U_ l = p_{i_ l}^{-1}(U_{l, i_ l})$ and $Z_ l = p_{j_ l}^{-1}(Z_{l, j_ l})$ with $U_{l, i_ l} \subset X_{i_ l}$ constructible open and $Z_{l, j_ l} \subset X_{j_ l}$ constructible closed. As $\mathcal{I}$ is cofiltered we may choose an object $k$ of $\mathcal{I}$ and morphism $a_ l : k \to i_ l$ and $b_ l : k \to j_ l$. Then taking $E_ k = \bigcup _{l = 1, \ldots , n} f_{a_ l}^{-1}(U_{l, i_ l}) \cap f_{b_ l}^{-1}(Z_{l, j_ l})$ we obtain a constructible subset of $X_ k$ whose inverse image in $X$ is $E$. $\square$

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Your email address will not be published.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 0A2Y. Beware of the difference between the letter 'O' and the digit '0'.