The Stacks project

Theorem 66.19.12. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp. surjective) if and only if the map on stalks $a_{\overline{x}} : \mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is injective (resp. surjective) for all geometric points of $X$. A sequence of abelian sheaves on $X_{\acute{e}tale}$ is exact if and only if it is exact on all stalks at geometric points of $S$.

Proof. We know the theorem is true if $X$ is a scheme, see Étale Cohomology, Theorem 59.29.10. Choose a surjective étale morphism $f : U \to X$ where $U$ is a scheme. Since $\{ U \to X\} $ is a covering (in $X_{spaces, {\acute{e}tale}}$) we can check whether a map of sheaves is injective, or surjective by restricting to $U$. Now if $\overline{u} : \mathop{\mathrm{Spec}}(k) \to U$ is a geometric point of $U$, then $(\mathcal{F}|_ U)_{\overline{u}} = \mathcal{F}_{\overline{x}}$ where $\overline{x} = f \circ \overline{u}$. (This is clear from the colimits defining the stalks at $\overline{u}$ and $\overline{x}$, but it also follows from Lemma 66.19.9.) Hence the result for $U$ implies the result for $X$ and we win. $\square$

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