For affine schemes, closed immersions correspond to ideals.

Lemma 26.8.2. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be an affine scheme. Let $i : Z \to X$ be any closed immersion of locally ringed spaces. Then there exists a unique ideal $I \subset R$ such that the morphism $i : Z \to X$ can be identified with the closed immersion $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$ constructed in Example 26.8.1 above.

Proof. This is kind of silly! Namely, by Lemma 26.4.5 we can identify $Z \to X$ with the closed subspace associated to a sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ as in Definition 26.4.4 and Example 26.4.3. By our conventions this sheaf of ideals is locally generated by sections as a sheaf of $\mathcal{O}_ X$-modules. Hence the quotient sheaf $\mathcal{O}_ X / \mathcal{I}$ is locally on $X$ the cokernel of a map $\bigoplus _{j \in J} \mathcal{O}_ U \to \mathcal{O}_ U$. Thus by definition, $\mathcal{O}_ X / \mathcal{I}$ is quasi-coherent. By our results in Section 26.7 it is of the form $\widetilde S$ for some $R$-module $S$. Moreover, since $\mathcal{O}_ X = \widetilde R \to \widetilde S$ is surjective we see by Lemma 26.7.8 that also $\mathcal{I}$ is quasi-coherent, say $\mathcal{I} = \widetilde I$. Of course $I \subset R$ and $S = R/I$ and everything is clear. $\square$

Comment #1105 by Evan Warner on

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