Definition 17.13.1. A *closed immersion of ringed spaces*^{1} is a morphism $i : (Z, \mathcal{O}_ Z) \to (X, \mathcal{O}_ X)$ with the following properties:

The map $i$ is a closed immersion of topological spaces.

The associated map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective. Denote the kernel by $\mathcal{I}$.

The $\mathcal{O}_ X$-module $\mathcal{I}$ is locally generated by sections.

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