Lemma 25.10.1. Let $X$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally ringed spaces.

1. The locally ringed space $Z$ is a scheme,

2. the kernel $\mathcal{I}$ of the map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is a quasi-coherent sheaf of ideals,

3. for any affine open $U = \mathop{\mathrm{Spec}}(R)$ of $X$ the morphism $i^{-1}(U) \to U$ can be identified with $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$ for some ideal $I \subset R$, and

4. we have $\mathcal{I}|_ U = \widetilde I$.

In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $X$ is a scheme.

Proof. Let $i : Z \to X$ be a closed immersion. Let $z \in Z$ be a point. Choose any affine open neighbourhood $i(z) \in U \subset X$. Say $U = \mathop{\mathrm{Spec}}(R)$. By Lemma 25.8.2 we know that $i^{-1}(U) \to U$ can be identified with the morphism of affine schemes $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$. First of all this implies that $z \in i^{-1}(U) \subset Z$ is an affine neighbourhood of $z$. Thus $Z$ is a scheme. Second this implies that $\mathcal{I}|_ U$ is $\widetilde I$. In other words for every point $x \in i(Z)$ there exists an open neighbourhood such that $\mathcal{I}$ is quasi-coherent in that neighbourhood. Note that $\mathcal{I}|_{X \setminus i(Z)} \cong \mathcal{O}_{X \setminus i(Z)}$. Thus the restriction of the sheaf of ideals is quasi-coherent on $X \setminus i(Z)$ also. We conclude that $\mathcal{I}$ is quasi-coherent. $\square$

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