Definition 35.8.2. Let $\tau \in \{ Zariski, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $S$ be a scheme. Let $\mathit{Sch}_\tau$ be a big site containing $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module.

1. The structure sheaf of the big site $(\mathit{Sch}/S)_\tau$ is the sheaf of rings $T/S \mapsto \Gamma (T, \mathcal{O}_ T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_ S$.

2. If $\tau = {\acute{e}tale}$ the structure sheaf of the small site $S_{\acute{e}tale}$ is the sheaf of rings $T/S \mapsto \Gamma (T, \mathcal{O}_ T)$ which is denoted $\mathcal{O}$ or $\mathcal{O}_ S$.

3. The sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$ on the big site $(\mathit{Sch}/S)_\tau$ is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma (T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^ a$ (and often simply $\mathcal{F}$).

4. Let $\tau = {\acute{e}tale}$ (resp. $\tau = Zariski$). The sheaf of $\mathcal{O}$-modules associated to $\mathcal{F}$ on the small site $S_{\acute{e}tale}$ (resp. $S_{Zar}$) is the sheaf of $\mathcal{O}$-modules $(f : T \to S) \mapsto \Gamma (T, f^*\mathcal{F})$ which is denoted $\mathcal{F}^ a$ (and often simply $\mathcal{F}$).

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