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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