Lemma 30.23.3. Let $X$ be a Noetherian scheme 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. Omitted. Hint: Look on affine opens, use Lemma 30.23.1, and use Algebra, Lemma 10.20.1. $\square$

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