Lemma 70.2.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $U \to X$ is an étale morphism such that $\text{WeakAss}(\mathcal{F}) \subset \mathop{\mathrm{Im}}(|U| \to |X|)$, then $\Gamma (X, \mathcal{F}) \to \Gamma (U, \mathcal{F})$ is injective.

**Proof.**
Let $s \in \Gamma (X, \mathcal{F})$ be a section which restricts to zero on $U$. Let $\mathcal{F}' \subset \mathcal{F}$ be the image of the map $\mathcal{O}_ X \to \mathcal{F}$ defined by $s$. Then $\mathcal{F}'|_ U = 0$. This implies that $\text{WeakAss}(\mathcal{F}') \cap \mathop{\mathrm{Im}}(|U| \to |X|) = \emptyset $ (by the definition of weakly associated points). On the other hand, $\text{WeakAss}(\mathcal{F}') \subset \text{WeakAss}(\mathcal{F})$ by Lemma 70.2.4. We conclude $\text{WeakAss}(\mathcal{F}') = \emptyset $. Hence $\mathcal{F}' = 0$ by Lemma 70.2.5.
$\square$

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