The Stacks project

Definition 66.18.1. Let $S$ be a scheme. Let $\mathit{Sch}_{fppf}$ be a big fppf site containing $S$, and let $\mathit{Sch}_{\acute{e}tale}$ be the corresponding big étale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The small étale site $X_{\acute{e}tale}$ of $X$ is defined as follows:

  1. An object of $X_{\acute{e}tale}$ is a morphism $\varphi : U \to X$ where $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale})$ is a scheme and $\varphi $ is an étale morphism,

  2. a morphism $(\varphi : U \to X) \to (\varphi ' : U' \to X)$ is given by a morphism of schemes $\chi : U \to U'$ such that $\varphi = \varphi ' \circ \chi $, and

  3. a family of morphisms $\{ (U_ i \to X) \to (U \to X)\} _{i \in I}$ of $X_{\acute{e}tale}$ is a covering if and only if $\{ U_ i \to U\} _{i \in I}$ is a covering of $(\mathit{Sch}/S)_{\acute{e}tale}$.

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Your email address will not be published.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



View Definition 66.18.1 as pdf