The Stacks project

Lemma 17.19.4. Let $X$ be a spectral topological space. Let $\mathcal{B}$ be the set of quasi-compact open subsets of $X$. Let $\mathcal{F}$ be a sheaf of sets as in Equation (17.19.2.1). Then there exist finitely many constructible closed subsets $Z_1, \ldots , Z_ n \subset X$ and finite sets $S_ i$ such that $\mathcal{F}$ is isomorphic to a subsheaf of $\prod (Z_ i \to X)_*\underline{S_ i}$.

Proof. By Lemma 17.19.3 we reduce to the case of a finite sober topological space and a sheaf with finite stalks. In this case $\mathcal{F} \subset \prod _{x \in X} i_{x, *}\mathcal{F}_ x$ where $i_ x : \{ x\} \to X$ is the embedding. We omit the proof that $i_{x, *}\mathcal{F}_ x$ is a constant sheaf on $\overline{\{ x\} }$. $\square$


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Your email address will not be published.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