The Stacks project

Lemma 34.4.13. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big ├ętale site containing $S$. The inclusion functor $S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$ satisfies the hypotheses of Sites, Lemma 7.21.8 and hence induces a morphism of sites

\[ \pi _ S : (\mathit{Sch}/S)_{\acute{e}tale}\longrightarrow S_{\acute{e}tale} \]

and a morphism of topoi

\[ i_ S : \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}) \]

such that $\pi _ S \circ i_ S = \text{id}$. Moreover, $i_ S = i_{\text{id}_ S}$ with $i_{\text{id}_ S}$ as in Lemma 34.4.12. In particular the functor $i_ S^{-1} = \pi _{S, *}$ is described by the rule $i_ S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.

Proof. In this case the functor $u : S_{\acute{e}tale}\to (\mathit{Sch}/S)_{\acute{e}tale}$, in addition to the properties seen in the proof of Lemma 34.4.12 above, also is fully faithful and transforms the final object into the final object. The lemma follows from Sites, Lemma 7.21.8. $\square$


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Your email address will not be published.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 021G. Beware of the difference between the letter 'O' and the digit '0'.