Lemma 10.21.2 (00ED)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.21: Open and closed subsets of spectra
Lemma 10.21.2 (cite)

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Lemma 10.21.2. Let $R_1$ and $R_2$ be rings. Let $R = R_1 \times R_2$ . The maps $R \to R_1$ , $(x, y) \mapsto x$ and $R \to R_2$ , $(x, y) \mapsto y$ induce continuous maps $\mathop{\mathrm{Spec}}(R_1) \to \mathop{\mathrm{Spec}}(R)$ and $\mathop{\mathrm{Spec}}(R_2) \to \mathop{\mathrm{Spec}}(R)$ . The induced map

$\mathop{\mathrm{Spec}}(R_1) \amalg \mathop{\mathrm{Spec}}(R_2) \longrightarrow \mathop{\mathrm{Spec}}(R)$

is a homeomorphism. In other words, the spectrum of $R = R_1\times R_2$ is the disjoint union of the spectrum of $R_1$ and the spectrum of $R_2$ .

Proof. Write $1 = e_1 + e_2$ with $e_1 = (1, 0)$ and $e_2 = (0, 1)$ . Note that $e_1$ and $e_2 = 1 - e_1$ are idempotents. We leave it to the reader to show that $R_1 = R_{e_1}$ is the localization of $R$ at $e_1$ . Similarly for $e_2$ . Thus the statement of the lemma follows from Lemma 10.21.1 combined with Lemma 10.17.6. $\square$

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