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The Stacks project

Lemma 59.29.7. Let S be a scheme. Let \overline{s} be a geometric point of S. Consider the functor

\begin{align*} u : S_{\acute{e}tale}& \longrightarrow \textit{Sets}, \\ U & \longmapsto |U_{\overline{s}}| = \{ \overline{u} \text{ such that }(U, \overline{u}) \text{ is an étale neighbourhood of }\overline{s}\} . \end{align*}

Here |U_{\overline{s}}| denotes the underlying set of the geometric fibre. Then u defines a point p of the site S_{\acute{e}tale} (Sites, Definition 7.32.2) and its associated stalk functor \mathcal{F} \mapsto \mathcal{F}_ p (Sites, Equation 7.32.1.1) is the functor \mathcal{F} \mapsto \mathcal{F}_{\overline{s}} defined above.

Proof. In the proof of Lemma 59.29.5 we have seen that the scheme U_{\overline{s}} is a disjoint union of schemes isomorphic to \overline{s}. Thus we can also think of |U_{\overline{s}}| as the set of geometric points of U lying over \overline{s}, i.e., as the collection of morphisms \overline{u} : \overline{s} \to U fitting into the diagram of Definition 59.29.1. From this it follows that u(S) is a singleton, and that u(U \times _ V W) = u(U) \times _{u(V)} u(W) whenever U \to V and W \to V are morphisms in S_{\acute{e}tale}. And, given a covering \{ U_ i \to U\} _{i \in I} in S_{\acute{e}tale} we see that \coprod u(U_ i) \to u(U) is surjective by Lemma 59.29.5. Hence Sites, Proposition 7.33.3 applies, so p is a point of the site S_{\acute{e}tale}. Finally, our functor \mathcal{F} \mapsto \mathcal{F}_{\overline{s}} is given by exactly the same colimit as the functor \mathcal{F} \mapsto \mathcal{F}_ p associated to p in Sites, Equation 7.32.1.1 which proves the final assertion. \square


Comments (2)

Comment #3237 by Dario Weißmann on

In the proof: Finally, the our functor

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  • 3 comment(s) on Section 59.29: Neighborhoods, stalks and points

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