Theorem 54.29.10. 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 54.29.9. 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. $\square$

Comment #1707 by Yogesh More on

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.

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