Proposition 59.70.3 (03S5)—The Stacks project

Proposition 59.70.3. Let $j : U \to X$ be an étale morphism of schemes. Let $\mathcal{F}$ in $\textit{Ab}(U_{\acute{e}tale})$. If $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ is a geometric point of $X$, then

\[ (j_!\mathcal{F})_{\overline{x}} = \bigoplus \nolimits _{\overline{u} : \mathop{\mathrm{Spec}}(k) \to U,\ j(\overline{u}) = \overline{x}} \mathcal{F}_{\bar{u}}. \]

In particular, $j_!$ is an exact functor.

Proof. Exactness of $j_!$ is very general, see Modules on Sites, Lemma 18.19.3. Of course it does also follow from the description of stalks. The formula for the stalk follows from Modules on Sites, Lemma 18.38.1 and the description of points of the small étale site in terms of geometric points, see Lemma 59.29.12.

For later use we note that the isomorphism

\begin{align*} (j_!\mathcal{F})_{\overline{x}} & = (j_{p!}\mathcal{F})_{\overline{x}} \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} j_{p!}\mathcal{F}(V) \\ & = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} \bigoplus \nolimits _{\varphi : V \to U} \mathcal{F}(V \xrightarrow {\varphi } U) \\ & \to \bigoplus \nolimits _{\overline{u} : \mathop{\mathrm{Spec}}(k) \to U,\ j(\overline{u}) = \overline{x}} \mathcal{F}_{\bar{u}}. \end{align*}

constructed in Modules on Sites, Lemma 18.38.1 sends $(V, \overline{v}, \varphi , s)$ to the class of $s$ in the stalk of $\mathcal{F}$ at $\overline{u} = \varphi (\overline{v})$. $\square$

The Stacks project

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