The Stacks project

Example 26.8.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Consider the morphism of affine schemes $i : Z = \mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R) = X$. By Algebra, Lemma 10.17.7 this is a homeomorphism of $Z$ onto a closed subset of $X$. Moreover, if $I \subset \mathfrak p \subset R$ is a prime corresponding to a point $x = i(z)$, $x \in X$, $z \in Z$, then on stalks we get the map

\[ \mathcal{O}_{X, x} = R_{\mathfrak p} \longrightarrow R_{\mathfrak p}/IR_{\mathfrak p} = \mathcal{O}_{Z, z} \]

Thus we see that $i$ is a closed immersion of locally ringed spaces, see Definition 26.4.1. Clearly, this is (isomorphic) to the closed subspace associated to the quasi-coherent sheaf of ideals $\widetilde I$, as in Example 26.4.3.

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