The Stacks project

Definition 61.12.8. Let $S$ be a scheme. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site containing $S$.

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

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

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


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