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Primes of a principal localization

Statement

Let $R$ be a commutative ring and let $f \in R$ . Contraction along $R \to R_{f}$ induces an inclusion-preserving bijection from $Spec (R_{f})$ to the set of prime ideals of $R$ that do not contain $f$ .

Facts & Assumptions

Given: A commutative ring $R$ and an element $f \in R$ .

[L1]

$R_{f}$ is the localization of $R$ at the multiplicative set ${1, f, f^{2}, \dots}$ (Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ ).

[L2]

Primes of a localization correspond to primes of the original ring disjoint from the denominator set (Primes of a localization avoid the denominator set).

Proof

technique · direct

1.1L1givenalgebra

By [L1], the denominator set is $S = {1, f, f^{2}, \dots}$ . A prime ideal $p$ is disjoint from $S$ exactly when $f \notin p$ : if $f \in p$ , then every positive power of $f$ lies in $p$ ; conversely, if some power of $f$ lies in $p$ , primality forces $f \in p$ .

2.1L2step 1.1

Applying [L2] to the localization at $S$ gives the stated bijection.

3.1step 2.1∎

Therefore primes of $R_{f}$ are exactly the primes of $R$ that avoid $f$ .

Depends on

Used by

Finite type need not be locally free Counterexample
The distinguished subset D(x) matches the primes of the localization at x Example

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 The spectrum of a ring (standard reference, not scraped)
The Stacks Project, Section 10.17: The spectrum of a ring (standard reference, not scraped)