Lemma 76.42.2. Let S be a scheme. Let X be a Noetherian algebraic space over S and let \mathcal{I} \subset \mathcal{O}_ X be a quasi-coherent sheaf of ideals. A map (\mathcal{F}_ n) \to (\mathcal{G}_ n) is surjective in \textit{Coh}(X, \mathcal{I}) if and only if \mathcal{F}_1 \to \mathcal{G}_1 is surjective.
Proof. We can check on an affine étale cover of X by Lemma 76.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.3. \square
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