Lemma 59.21.2. Let $S$ be a scheme. The étale topos of $S$ is independent (up to canonical equivalence) of the construction of the small étale site in Definition 59.20.2.

Proof. We have to show, given two big étale sites $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$ containing $S$, then $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \cong \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}')$ with obvious notation. By Topologies, Lemma 34.12.1 we may assume $\mathit{Sch}_{\acute{e}tale}\subset \mathit{Sch}_{\acute{e}tale}'$. By Sets, Lemma 3.9.9 any affine scheme étale over $S$ is isomorphic to an object of both $\mathit{Sch}_{\acute{e}tale}$ and $\mathit{Sch}_{\acute{e}tale}'$. Thus the induced functor $S_{affine, {\acute{e}tale}} \to S_{affine, {\acute{e}tale}}'$ is an equivalence. Moreover, it is clear that both this functor and a quasi-inverse map transform standard étale coverings into standard étale coverings. Hence the result follows from Topologies, Lemma 34.4.12. $\square$

## 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Your email address will not be published.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 0958. Beware of the difference between the letter 'O' and the digit '0'.