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The Stacks project

Remarks 59.29.11. On points of the geometric sites.

  1. Theorem 59.29.10 says that the family of points of S_{\acute{e}tale} given by the geometric points of S (Lemma 59.29.7) is conservative, see Sites, Definition 7.38.1. In particular S_{\acute{e}tale} has enough points.

  2. Suppose \mathcal{F} is a sheaf on the big étale site of S. Let T \to S be an object of the big étale site of S, and let \overline{t} be a geometric point of T. Then we define \mathcal{F}_{\overline{t}} as the stalk of the restriction \mathcal{F}|_{T_{\acute{e}tale}} of \mathcal{F} to the small étale site of T. In other words, we can define the stalk of \mathcal{F} at any geometric point of any scheme T/S \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{\acute{e}tale}).

  3. The big étale site of S also has enough points, by considering all geometric points of all objects of this site, see (2).


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