# The Stacks Project

## Tag 03PU

Theorem 53.29.10. Let $S$ be a scheme. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp. surjective) if and only if the map on stalks $a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is injective (resp. surjective) for all geometric points of $S$. A sequence of abelian sheaves on $S_{\acute{e}tale}$ is exact if and only if it is exact on all stalks at geometric points of $S$.

Proof. The necessity of exactness on stalks follows from Lemma 53.29.9. For the converse, it suffices to show that a map of sheaves is surjective (respectively injective) if and only if it is surjective (respectively injective) on all stalks. We prove this in the case of surjectivity, and omit the proof in the case of injectivity.

Let $\alpha : \mathcal{F} \to \mathcal{G}$ be a map of sheaves such that $\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is surjective for all geometric points. Fix $U \in \mathop{\mathrm{Ob}}\nolimits(S_{\acute{e}tale})$ and $s \in \mathcal{G}(U)$. For every $u \in U$ choose some $\overline{u} \to U$ lying over $u$ and an étale neighborhood $(V_u , \overline{v}_u) \to (U, \overline{u})$ such that $s|_{V_u} = \alpha(s_{V_u})$ for some $s_{V_u} \in \mathcal{F}(V_u)$. This is possible since $\alpha$ is surjective on stalks. Then $\{V_u \to U\}_{u \in U}$ is an étale covering on which the restrictions of $s$ are in the image of the map $\alpha$. Thus, $\alpha$ is surjective, see Sites, Section 7.11. $\square$

The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 3512–3521 (see updates for more information).

\begin{theorem}
\label{theorem-exactness-stalks}
Let $S$ be a scheme.
A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective
(resp.\ surjective) if and only if the map on stalks
$a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$
is injective (resp.\ surjective) for all geometric points of $S$.
A sequence of abelian sheaves on $S_\etale$ is exact
if and only if it is exact on all stalks at geometric points of $S$.
\end{theorem}

\begin{proof}
The necessity of exactness on stalks follows from
Lemma \ref{lemma-stalk-exact}.
For the converse, it suffices to show that a map of sheaves is surjective
(respectively injective) if and only if it is surjective (respectively
injective) on all stalks. We prove this in the case of surjectivity, and omit
the proof in the case of injectivity.

\medskip\noindent
Let $\alpha : \mathcal{F} \to \mathcal{G}$ be a map of sheaves such
that $\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$
is surjective for all geometric points. Fix
$U \in \Ob(S_\etale)$
and $s \in \mathcal{G}(U)$. For every $u \in U$ choose some
$\overline{u} \to U$ lying over $u$ and an \'etale neighborhood
$(V_u , \overline{v}_u) \to (U, \overline{u})$ such that
$s|_{V_u} = \alpha(s_{V_u})$ for some
$s_{V_u} \in \mathcal{F}(V_u)$.
This is possible since $\alpha$ is surjective on
stalks. Then $\{V_u \to U\}_{u \in U}$
is an \'etale covering on which the restrictions of $s$
are in the image of the map $\alpha$.
Thus, $\alpha$ is surjective, see
Sites, Section \ref{sites-section-sheaves-injective}.
\end{proof}

Comment #1707 by Yogesh More on November 30, 2015 a 12:24 pm UTC

very minor remark: at the beginning of the second paragraph of the proof, "Let $\alpha:F \to G$ be a map of abelian sheaves", I don't think you need the word abelian; the statement of the theorem says it holds for sheaves of sets.

Comment #1752 by Johan (site) on December 15, 2015 a 7:02 pm UTC

THanks, fixed here.

There are also 2 comments on Section 53.29: Étale Cohomology.

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