The Stacks project

Definition 34.3.7. Let $S$ be a scheme. Let $\mathit{Sch}_{Zar}$ be a big Zariski site containing $S$.

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

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

3. The big affine Zariski site of $S$, denoted $(\textit{Aff}/S)_{Zar}$, is the full subcategory of $(\mathit{Sch}/S)_{Zar}$ consisting of objects $U/S$ such that $U$ is an affine scheme. A covering of $(\textit{Aff}/S)_{Zar}$ is any covering $\{ U_ i \to U\}$ of $(\mathit{Sch}/S)_{Zar}$ with $U \in \mathop{\mathrm{Ob}}\nolimits ((\textit{Aff}/S)_{Zar})$ which is a standard Zariski covering.

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