Definition 26.4.1. Let $i : Z \to X$ be a morphism of locally ringed spaces. We say that $i$ is a closed immersion if:

1. The map $i$ is a homeomorphism of $Z$ onto a closed subset of $X$.

2. The map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective; let $\mathcal{I}$ denote the kernel.

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

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