Lemma 71.2.9. 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 71.2.4. We conclude \text{WeakAss}(\mathcal{F}') = \emptyset . Hence \mathcal{F}' = 0 by Lemma 71.2.5. \square
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