The Stacks project

Remark 59.23.2. Let $G$ be an abstract group. On any of the sites $(\mathit{Sch}/S)_\tau $ or $S_\tau $ of Section 59.20 the sheafification $\underline{G}$ of the constant presheaf associated to $G$ in the Zariski topology of the site already gives

\[ \Gamma (U, \underline{G}) = \{ \text{Zariski locally constant maps }U \to G\} \]

This Zariski sheaf is representable by the group scheme $G_ S$ according to Groupoids, Example 39.5.6. By Lemma 59.15.8 any representable presheaf satisfies the sheaf condition for the $\tau $-topology as well, and hence we conclude that the Zariski sheafification $\underline{G}$ above is also the $\tau $-sheafification.

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Your email address will not be published.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 03P5. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 59.23.2 as pdf