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 those $U/S$ such that $U$ is an affine scheme. A covering of $(\textit{Aff}/S)_{\acute{e}tale}$ is any covering $\{ U_ i \to U\} $ of $(\mathit{Sch}/S)_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits ((\textit{Aff}/S)_{\acute{e}tale})$ which is a standard étale covering.

  4. The small affine étale site of $S$, denoted $S_{affine, {\acute{e}tale}}$, is the full subcategory of $S_{\acute{e}tale}$ whose objects are those $U/S$ such that $U$ is an affine scheme. A covering of $S_{affine, {\acute{e}tale}}$ is any covering $\{ U_ i \to U\} $ of $S_{\acute{e}tale}$ with $U \in \mathop{\mathrm{Ob}}\nolimits (S_{affine, {\acute{e}tale}})$ which is a standard étale covering.

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