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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