The Stacks project

Proof. The necessity of exactness on stalks follows from Lemma 59.29.9. 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. $\square$