Lemma 59.30.1 (06VX)—The Stacks project

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 $S_{affine, {\acute{e}tale}}$. $\square$

Comments (3)

Comment #2585 by Ingo Blechschmidt on May 31, 2017 at 00:40

Deligne's referenced result requires that the site contains finite limits. But does $\mathrm{Aff}/S$ contain a terminal object? I'm under the impression that $\mathrm{Aff}/S$ is the category of $S$ -schemes which are affine as schemes over $\operatorname{Spec} \mathbb{Z}$ , not the category of $S$ -schemes whose structural morphism to $S$ is affine.

Comment #2586 by Johan on May 31, 2017 at 14:01

Yes, that is a mistake. Thanks very much. The point is that it locally has the right structure. I've fixed this here.

Comment #2587 by Ingo Blechschmidt on June 01, 2017 at 15:41

Perfect, that covers it. Thank you!

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