The Stacks project

Definition 34.4.8. Let $S$ be a scheme. Let $\mathit{Sch}_{\acute{e}tale}$ be a big étale site containing $S$.

  1. The big étale site of $S$, denoted $(\mathit{Sch}/S)_{\acute{e}tale}$, is the site $\mathit{Sch}_{\acute{e}tale}/S$ introduced in Sites, Section 7.25.

  2. The small étale site of $S$, which we denote $S_{\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ whose objects are those $U/S$ such that $U \to S$ is étale. A covering of $S_{\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{\acute{e}tale})$.

  3. The big affine étale site of $S$, denoted $(\textit{Aff}/S)_{\acute{e}tale}$, is the full subcategory of $(\mathit{Sch}/S)_{\acute{e}tale}$ whose objects are affine $U/S$. A covering of $(\textit{Aff}/S)_{\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ which is a standard étale covering.


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