**Proof.**
Before we indicate how to prove this by direct arguments we note that the result follows from the general material in Modules on Sites, Section 18.36. This is true because $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ comes from a point of the small étale site of $S$, see Lemma 59.29.7. We will only give a direct proof of (1), (2) and (3), and omit a direct proof of (4).

Exactness as a functor on $\textit{PAb}(S_{\acute{e}tale})$ is formal from the fact that directed colimits commute with all colimits and with finite limits. The identification of the stalks in (2) is via the map

\[ \kappa : \mathcal{F}_{\overline{s}} \longrightarrow (\mathcal{F}^\# )_{\overline{s}} \]

induced by the natural morphism $\mathcal{F}\to \mathcal{F}^\# $, see Theorem 59.13.2. We claim that this map is an isomorphism of abelian groups. We will show injectivity and omit the proof of surjectivity.

Let $\sigma \in \mathcal{F}_{\overline{s}}$. There exists an étale neighborhood $(U, \overline{u})\to (S, \overline{s})$ such that $\sigma $ is the image of some section $s \in \mathcal{F}(U)$. If $\kappa (\sigma ) = 0$ in $(\mathcal{F}^\# )_{\overline{s}}$ then there exists a morphism of étale neighborhoods $(U', \overline{u}')\to (U, \overline{u})$ such that $s|_{U'}$ is zero in $\mathcal{F}^\# (U')$. It follows there exists an étale covering $\{ U_ i'\to U'\} _{i\in I}$ such that $s|_{U_ i'}=0$ in $\mathcal{F}(U_ i')$ for all $i$. By Lemma 59.29.5 there exist $i \in I$ and a morphism $\overline{u}_ i': \overline{s} \to U_ i'$ such that $(U_ i', \overline{u}_ i') \to (U', \overline{u}')\to (U, \overline{u})$ are morphisms of étale neighborhoods. Hence $\sigma = 0$ since $(U_ i', \overline{u}_ i') \to (U, \overline{u})$ is a morphism of étale neighbourhoods such that we have $s|_{U'_ i}=0$. This proves $\kappa $ is injective.

To show that the functor $\textit{Ab}(S_{\acute{e}tale}) \to \textit{Ab}$ is exact, consider any short exact sequence in $\textit{Ab}(S_{\acute{e}tale})$: $ 0\to \mathcal{F}\to \mathcal{G}\to \mathcal H \to 0. $ This gives us the exact sequence of presheaves

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal H \to \mathcal H/^ p\mathcal{G} \to 0, \]

where $/^ p$ denotes the quotient in $\textit{PAb}(S_{\acute{e}tale})$. Taking stalks at $\overline{s}$, we see that $(\mathcal H /^ p\mathcal{G})_{\bar{s}} = (\mathcal H /\mathcal{G})_{\bar{s}} = 0$, since the sheafification of $\mathcal H/^ p\mathcal{G}$ is $0$. Therefore,

\[ 0\to \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}} \to \mathcal{H}_{\overline{s}} \to 0 = (\mathcal H/^ p\mathcal{G})_{\overline{s}} \]

is exact, since taking stalks is exact as a functor from presheaves.
$\square$

## Comments (2)

Comment #8268 by Xiaolong Liu on

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