The Stacks project

Lemma 61.6.1. Let $A$ be a ring. Let $X = \mathop{\mathrm{Spec}}(A)$. Let $T \subset \pi _0(X)$ be a closed subset. There exists a surjective ind-Zariski ring map $A \to B$ such that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ induces a homeomorphism of $\mathop{\mathrm{Spec}}(B)$ with the inverse image of $T$ in $X$.

Proof. Let $Z \subset X$ be the inverse image of $T$. Then $Z$ is the intersection $Z = \bigcap Z_\alpha $ of the open and closed subsets of $X$ containing $Z$, see Topology, Lemma 5.12.12. For each $\alpha $ we have $Z_\alpha = \mathop{\mathrm{Spec}}(A_\alpha )$ where $A \to A_\alpha $ is a local isomorphism (a localization at an idempotent). Setting $B = \mathop{\mathrm{colim}}\nolimits A_\alpha $ proves the lemma. $\square$

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