Lemma 29.2.3. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a sheaf of ideals. The following are equivalent:

$\mathcal{I}$ is locally generated by sections as a sheaf of $\mathcal{O}_ X$-modules,

$\mathcal{I}$ is quasi-coherent as a sheaf of $\mathcal{O}_ X$-modules, and

there exists a closed immersion $i : Z \to X$ of schemes whose corresponding sheaf of ideals $\mathop{\mathrm{Ker}}(i^\sharp )$ is equal to $\mathcal{I}$.

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