Definition 17.13.1. A closed immersion of ringed spaces1 is a morphism $i : (Z, \mathcal{O}_ Z) \to (X, \mathcal{O}_ X)$ with the following properties:

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

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

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

[1] This is nonstandard notation; see discussion above.

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