Lemma 29.2.1. Let $i : Z \to X$ be a morphism of schemes. The following are equivalent:

The morphism $i$ is a closed immersion.

For every affine open $\mathop{\mathrm{Spec}}(R) = U \subset X$, there exists an ideal $I \subset R$ such that $i^{-1}(U) = \mathop{\mathrm{Spec}}(R/I)$ as schemes over $U = \mathop{\mathrm{Spec}}(R)$.

There exists an affine open covering $X = \bigcup _{j \in J} U_ j$, $U_ j = \mathop{\mathrm{Spec}}(R_ j)$ and for every $j \in J$ there exists an ideal $I_ j \subset R_ j$ such that $i^{-1}(U_ j) = \mathop{\mathrm{Spec}}(R_ j/I_ j)$ as schemes over $U_ j = \mathop{\mathrm{Spec}}(R_ j)$.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$ and $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective, and the kernel $\mathop{\mathrm{Ker}}(i^\sharp )\subset \mathcal{O}_ X$ is a quasi-coherent sheaf of ideals.

The morphism $i$ induces a homeomorphism of $Z$ with a closed subset of $X$, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective, and the kernel $\mathop{\mathrm{Ker}}(i^\sharp )\subset \mathcal{O}_ X$ is a sheaf of ideals which is locally generated by sections.

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