The Stacks project

Lemma 98.25.2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Restricting along the inclusion functor $(\textit{Noetherian}/S)_\tau \to (\mathit{Sch}/S)_\tau $ defines an equivalence of categories between

  1. the category of limit preserving sheaves on $(\mathit{Sch}/S)_\tau $ and

  2. the category of limit preserving sheaves on $(\textit{Noetherian}/S)_\tau $

Proof. Let $F : (\textit{Noetherian}/S)_\tau ^{opp} \to \textit{Sets}$ be a functor which is both limit preserving and a sheaf. By Topologies, Lemmas 34.13.1 and 34.13.3 there exists a unique functor $F' : (\mathit{Sch}/S)_\tau ^{opp} \to \textit{Sets}$ which is limit preserving, a sheaf, and restricts to $F$. In fact, the construction of $F'$ in Topologies, Lemma 34.13.1 is functorial in $F$ and this construction is a quasi-inverse to restriction. Some details omitted. $\square$


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Your email address will not be published.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 0GE3. Beware of the difference between the letter 'O' and the digit '0'.