Lemma 74.42.2. 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 74.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.3.
$\square$

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