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 $(\mathit{Sch}/S)_{\acute{e}tale}$.

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