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The spectrum of a principal localisation is the distinguished open D(f)

Statement

Let $R$ be a commutative ring and let $f \in R$ . The localisation map $R \to R_{f}$ induces a homeomorphism from $Spec (R_{f})$ onto the distinguished open subset $D (f) = {p \in Spec (R) : f \notin p} .$

Facts & Assumptions

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

[L1]

For a localization at a multiplicative set $S$ , the spectrum identifies homeomorphically with the primes of $R$ disjoint from $S$ (The spectrum of a localisation is the subspace of primes disjoint from the denominator set).

[L2]

$D (f)$ is the set of prime ideals of $R$ that do not contain $f$ (Principal distinguished subsets of the prime spectrum).

Proof

technique · direct

1.1L1

Apply [L1] with $S = {1, f, f^{2}, \dots}$ . Its image consists of the prime ideals $p$ such that $p \cap S = \emptyset$ .

1.2L2givenalgebra

For a prime ideal $p$ , the condition $p \cap {1, f, f^{2}, \dots} = \emptyset$ is equivalent to $f \notin p$ , because $f \in p$ implies $f^{n} \in p$ for every $n \geq 1$ , while $f^{n} \in p$ implies $f \in p$ by primality. By [L2], this image is exactly $D (f)$ .

2.1step 1.1step 1.2∎

Therefore $Spec (R_{f})$ is homeomorphic to the distinguished open subset $D (f)$ .

Depends on

Used by

Every distinguished open subset is compact Corollary
Finite type need not be locally free Counterexample
Two-affine projective line and its twists Definition
Open immersions are etale Example
The punctured affine line as an open finite factorization Example
A dominant map has a surjective differential on a dense source open Lemma
A principal localization identifies its spectrum with a distinguished open Lemma
Constructible subsets stable under generalisation are open in an affine spectrum Lemma
Affine acyclicity of quasi-coherent sheaves Theorem

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 (standard reference, not scraped)
The Stacks Project, Lemma 10.17.6 (standard reference, not scraped)