The Stacks project

Lemma 66.19.13. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ be a point of the small étale topos of $X$. Then there exists a geometric point $\overline{x}$ of $X$ such that the stalk functor $\mathcal{F} \mapsto \mathcal{F}_ p$ is isomorphic to the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$.

Proof. By Sites, Lemma 7.32.7 there is a one to one correspondence between points of the site and points of the associated topos. Hence we may assume that $p$ is given by a functor $u : X_{\acute{e}tale}\to \textit{Sets}$ which defines a point of the site $X_{\acute{e}tale}$. Let $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ be an object whose structure morphism $j : U \to X$ is surjective. Note that $h_ U$ is a sheaf which surjects onto the final sheaf. Since taking stalks is exact we see that $(h_ U)_ p = u(U)$ is not empty (use Sites, Lemma 7.32.3). Pick $x \in u(U)$. By Sites, Lemma 7.35.1 we obtain a point $q : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ such that $p = j_{small} \circ q$, so that $\mathcal{F}_ p = (\mathcal{F}|_ U)_ q$ functorially. By Étale Cohomology, Lemma 59.29.12 there is a geometric point $\overline{u}$ of $U$ and a functorial isomorphism $\mathcal{G}_ q = \mathcal{G}_{\overline{u}}$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$. Set $\overline{x} = j \circ \overline{u}$. Then we see that $\mathcal{F}_{\overline{x}} \cong (\mathcal{F}|_ U)_{\overline{u}}$ functorially in $\mathcal{F}$ on $X_{\acute{e}tale}$ by Lemma 66.19.9 and we win. $\square$

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