The Stacks project

Lemma 59.29.9. Let $S$ be a scheme. Let $\overline{s}$ be a geometric point of $S$.

  1. The stalk functor $\textit{PAb}(S_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact.

  2. We have $(\mathcal{F}^\# )_{\overline{s}} = \mathcal{F}_{\overline{s}}$ for any presheaf of sets $\mathcal{F}$ on $S_{\acute{e}tale}$.

  3. The functor $\textit{Ab}(S_{\acute{e}tale}) \to \textit{Ab}$, $\mathcal{F} \mapsto \mathcal{F}_{\overline{s}}$ is exact.

  4. Similarly the functors $\textit{PSh}(S_{\acute{e}tale}) \to \textit{Sets}$ and $\mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale}) \to \textit{Sets}$ given by the stalk functor $\mathcal{F} \mapsto \mathcal{F}_{\overline{x}}$ are exact (see Categories, Definition 4.23.1) and commute with arbitrary colimits.

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. $\square$

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