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.

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 $\mathcal{O}_ S$.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 $\mathcal{O}_ S$.

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 $\mathcal{F}$).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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