The Stacks project

Lemma 34.4.9. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$. The structures $S_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, and $S_{affine, {\acute{e}tale}}$ are sites.

Proof. Let us show that $S_{\acute{e}tale}$ is a site. It is a category with a given set of families of morphisms with fixed target. Thus we have to show properties (1), (2) and (3) of Sites, Definition 7.6.2. Since $(\mathit{Sch}/S)_{\acute{e}tale}$ is a site, it suffices to prove that given any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$ we also have $U_ i \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$. This follows from the definitions as the composition of étale morphisms is an étale morphism.

Let us show that $(\textit{Aff}/S)_{\acute{e}tale}$ is a site. Reasoning as above, it suffices to show that the collection of standard étale coverings of affines satisfies properties (1), (2) and (3) of Sites, Definition 7.6.2. This is clear since for example, given a standard étale covering $\{ T_ i \to T\} _{i\in I}$ and for each $i$ we have a standard étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a standard étale covering because $\bigcup _{i\in I} J_ i$ is finite and each $T_{ij}$ is affine.

We omit the proof that $S_{affine, étale}$ is a site. $\square$

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Your email address will not be published.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