Loading web-font TeX/Caligraphic/Regular

The Stacks project

Lemma 5.24.2. 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.

  1. Given nonempty subsets Z_ i \subset X_ i closed in the constructible topology with f_ a(Z_ j) \subset Z_ i for all a : j \to i in \mathcal{I}, then \mathop{\mathrm{lim}}\nolimits Z_ i is nonempty.

  2. If each X_ i is nonempty, then X = \mathop{\mathrm{lim}}\nolimits X_ i is nonempty.

Proof. Denote X'_ i the space X_ i endowed with the constructible topology and Z'_ i the corresponding subspace of X'_ i. Let a : j \to i in \mathcal{I} be a morphism. As f_ a is spectral it defines a continuous map f_ a : X'_ j \to X'_ i. Thus f_ a|_{Z_ j} : Z'_ j \to Z'_ i is a continuous map of quasi-compact Hausdorff spaces (by Lemmas 5.23.2 and 5.12.3). By Lemma 5.14.6 the space \mathop{\mathrm{lim}}\nolimits Z'_ i is nonempty. Since \mathop{\mathrm{lim}}\nolimits Z'_ i = \mathop{\mathrm{lim}}\nolimits Z_ i as sets we conclude. \square


Comments (0)


Your email address will not be published.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.