The Stacks project

Example 26.4.3. Let $X$ be a locally ringed space. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a sheaf of ideals which is locally generated by sections as a sheaf of $\mathcal{O}_ X$-modules. Let $Z$ be the support of the sheaf of rings $\mathcal{O}_ X/\mathcal{I}$. This is a closed subset of $X$, by Modules, Lemma 17.5.3. Denote $i : Z \to X$ the inclusion map. By Modules, Lemma 17.6.1 there is a unique sheaf of rings $\mathcal{O}_ Z$ on $Z$ with $i_*\mathcal{O}_ Z = \mathcal{O}_ X/\mathcal{I}$. For any $z \in Z$ the stalk $\mathcal{O}_{Z, z}$ is equal to a quotient $\mathcal{O}_{X, i(z)}/\mathcal{I}_{i(z)}$ of a local ring and nonzero, hence a local ring. Thus $i : (Z, \mathcal{O}_ Z) \to (X, \mathcal{O}_ X)$ is a closed immersion of locally ringed spaces.

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