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The Stacks project

Lemma 71.2.5. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Then

\mathcal{F} = (0) \Leftrightarrow \text{WeakAss}(\mathcal{F}) = \emptyset

Proof. Choose a scheme U and a surjective étale morphism f : U \to X. Then \mathcal{F} is zero if and only if f^*\mathcal{F} is zero. Hence the lemma follows from the definition and the lemma in the case of schemes, see Divisors, Lemma 31.5.5. \square


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