The Stacks project

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:

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

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

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

Proof. The equivalence of (1) and (2) is immediate from Schemes, Lemma 26.10.1. If (1) holds, then there is a closed subspace $i : Z \to X$ with $\mathcal{I} = \mathop{\mathrm{Ker}}(i^\sharp )$ by Schemes, Definition 26.4.4 and Example 26.4.3. By Schemes, Lemma 26.10.1 this is a closed immersion of schemes and (3) holds. Conversely, if (3) holds, then (2) holds by Schemes, Lemma 26.10.1 (which applies because a closed immersion of schemes is a fortiori a closed immersion of locally ringed spaces). $\square$

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