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. $\square$

Comment #3237 by Dario Weißmann on

In the proof: Finally, the our functor

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