The Stacks project

Lemma 96.9.5. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. The site $\mathcal{X}_\tau $ has enough points.

Proof. By Sites, Lemma 7.38.5 we have to show that there exists a family of objects $x$ of $\mathcal{X}$ such that $\mathcal{X}_\tau /x$ has enough points and such that the sheaves $h_ x^\# $ cover the final object of the category of sheaves. By Lemma 96.9.1 and Étale Cohomology, Lemma 59.30.1 we see that $\mathcal{X}_\tau /x$ has enough points for every object $x$ and we win. $\square$

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