Lemma 10.17.7 (00E5)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.17: The spectrum of a ring
Lemma 10.17.7 (cite)

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Lemma 10.17.7. Let $R$ be a ring. Let $I \subset R$ be an ideal. The map $R \to R/I$ induces via the functoriality of $\mathop{\mathrm{Spec}}$ a homeomorphism

$\mathop{\mathrm{Spec}}(R/I) \longrightarrow V(I) \subset \mathop{\mathrm{Spec}}(R).$

The inverse is given by $\mathfrak p \mapsto \mathfrak p / I$ .

Proof. It is immediate that the image is contained in $V(I)$ . On the other hand, if $\mathfrak p \in V(I)$ then $\mathfrak p \supset I$ and we may consider the ideal $\mathfrak p /I \subset R/I$ . Using basic notion (51) we see that $(R/I)/(\mathfrak p/I) = R/\mathfrak p$ is a domain and hence $\mathfrak p/I$ is a prime ideal. From this it is immediately clear that the image of $D(f + I)$ is $D(f) \cap V(I)$ , and hence the map is a homeomorphism. $\square$

Comments (1)

Comment #626 by Wei Xu on May 30, 2014 at 06:28

A typo: In the statement, "homeorphism" should be "homeomorphism".

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