Lemma 10.41.7 (00I0)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.41: Going up and going down
Lemma 10.41.7 (cite)

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Lemma 10.41.7. Let $R$ be a ring. Let $E \subset \mathop{\mathrm{Spec}}(R)$ be a constructible subset.

If $E$ is stable under specialization, then $E$ is closed.
If $E$ is stable under generalization, then $E$ is open.

Proof. First proof. The first assertion follows from Lemma 10.41.5 combined with Lemma 10.29.4. The second follows because the complement of a constructible set is constructible (see Topology, Lemma 5.15.2), the first part of the lemma and Topology, Lemma 5.19.2.

Second proof. Since $\mathop{\mathrm{Spec}}(R)$ is a spectral space by Lemma 10.26.2 this is a special case of Topology, Lemma 5.23.6. $\square$

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