The Stacks project

Remark 59.56.6. Let $S$ be a scheme and let $\overline{s} : \mathop{\mathrm{Spec}}(k) \to S$ be a geometric point of $S$. By definition this means that $k$ is algebraically closed. In particular the absolute Galois group of $k$ is trivial. Hence by Theorem 59.56.3 the category of sheaves on $\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}$ is equivalent to the category of sets. The equivalence is given by taking sections over $\mathop{\mathrm{Spec}}(k)$. This finally provides us with an alternative definition of the stalk functor. Namely, the functor

\[ \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \longrightarrow \textit{Sets}, \quad \mathcal{F} \longmapsto \mathcal{F}_{\overline{s}} \]

is isomorphic to the functor

\[ \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathop{\mathrm{Spec}}(k)_{\acute{e}tale}) = \textit{Sets}, \quad \mathcal{F} \longmapsto \overline{s}^*\mathcal{F} \]

To prove this rigorously one can use Lemma 59.36.2 part (3) with $f = \overline{s}$. Moreover, having said this the general case of Lemma 59.36.2 part (3) follows from functoriality of pullbacks.

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