Lemma 71.2.4. Let S be a scheme. Let X be an algebraic space over S. Let 0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0 be a short exact sequence of quasi-coherent sheaves on X. Then \text{WeakAss}(\mathcal{F}_2) \subset \text{WeakAss}(\mathcal{F}_1) \cup \text{WeakAss}(\mathcal{F}_3) and \text{WeakAss}(\mathcal{F}_1) \subset \text{WeakAss}(\mathcal{F}_2).
Proof. For every geometric point \overline{x} \in X the sequence of stalks 0 \to \mathcal{F}_{1, \overline{x}} \to \mathcal{F}_{2, \overline{x}} \to \mathcal{F}_{3, \overline{x}} \to 0 is a short exact sequence of \mathcal{O}_{X, \overline{x}}-modules. Hence the lemma follows from Algebra, Lemma 10.66.4. \square
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