The Stacks project

Proof. Let $U \subset X$ be an open subset and let $s \in \mathcal{J}(U)$ be a section. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals defining the reduced induced scheme structure on $X \setminus U$ (see Schemes, Definition 26.12.5). By Cohomology of Schemes, Lemma 30.10.5 the section $s$ corresponds to a map $\sigma : \mathcal{I}^ n \to \mathcal{J}$ for some $n$. As $\mathcal{J}$ is an injective object of $\mathit{QCoh}(\mathcal{O}_ X)$ we can extend $\sigma $ to a map $\tilde s : \mathcal{O}_ X \to \mathcal{J}$. Then $\tilde s$ corresponds to a global section of $\mathcal{J}$ restricting to $s$. $\square$