Lemma 59.30.1. Let $S$ be a scheme. All of the following sites have enough points $S_{affine, Zar}$, $S_{Zar}$, $S_{affine, {\acute{e}tale}}$, $S_{\acute{e}tale}$, $(\mathit{Sch}/S)_{Zar}$, $(\textit{Aff}/S)_{Zar}$, $(\mathit{Sch}/S)_{\acute{e}tale}$, $(\textit{Aff}/S)_{\acute{e}tale}$, $(\mathit{Sch}/S)_{smooth}$, $(\textit{Aff}/S)_{smooth}$, $(\mathit{Sch}/S)_{syntomic}$, $(\textit{Aff}/S)_{syntomic}$, $(\mathit{Sch}/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$.

**Proof.**
For each of the big sites the associated topos is equivalent to the topos defined by the site $(\textit{Aff}/S)_\tau $, see Topologies, Lemmas 34.3.10, 34.4.11, 34.5.9, 34.6.9, and 34.7.11. The result for the sites $(\textit{Aff}/S)_\tau $ follows immediately from Deligne's result Sites, Lemma 7.39.4.

The result for $S_{Zar}$ is clear. The result for $S_{affine, Zar}$ follows from Deligne's result. The result for $S_{\acute{e}tale}$ either follows from (the proof of) Theorem 59.29.10 or from Topologies, Lemma 34.4.12 and Deligne's result applied to $S_{affine, {\acute{e}tale}}$. $\square$

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