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LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05
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The spectrum of a localisation is the subspace of primes disjoint from the denominator set

Statement

Let R be a commutative ring, let S⊆R be multiplicative, and let λ:R→S−1R be the localisation map. Then contraction along λ is a homeomorphism from Spec⁡(S−1R) onto the subspace X:={p∈Spec⁡(R):p∩S=∅}.

Facts & Assumptions

Given: A commutative ring R, a multiplicative subset S⊆R, and the localization map λ:R→S−1R.

[L1]

Contraction along λ is an inclusion-preserving bijection from Spec⁡(S−1R) onto X, with inverse p↦S−1p (Prime ideals of a localization are exactly the primes disjoint from the denominator set).

[L2]

In a Zariski spectrum, the closed sets are precisely the vanishing sets (The vanishing sets define the Zariski topology on the prime spectrum).

[A1]

If K⊴S−1R and I=λ−1(K), then K=S−1I.

[A2]

A subset of X is closed in the subspace topology exactly when it has the form X∩V(I) for some ideal I⊴R.

Proof

technique · direct
1.1L1

By [L1], contraction gives a bijection c:Spec⁡(S−1R)→X.

1.2L1givenalgebra

Let I⊴R. If q∈Spec⁡(S−1R) has contraction p, then the inverse description in [L1] gives q=S−1p. Therefore q⊇S−1I⟺p⊇I, and hence c(VS−1R(S−1I))=X∩VR(I).

2.1L2A1step 1.2

By [L2], every closed subset of Spec⁡(S−1R) has the form VS−1R(K) for some ideal K⊴S−1R. With I=λ−1(K), assumption [A1] gives K=S−1I, so step 1.2 shows that c sends every closed subset of Spec⁡(S−1R) to a closed subset of X.

2.2A2step 1.2

Conversely, if C⊆X is closed, then [A2] gives C=X∩VR(I) for some ideal I⊴R. Step 1.2 then yields C=c(VS−1R(S−1I)), so c−1 also preserves closed sets.

3.1step 1.1step 2.1step 2.2∎

The bijection c and its inverse both preserve closed sets, so c is a homeomorphism from Spec⁡(S−1R) onto the subspace X.

Depends on

Used by

Dependency tree · two levels

6 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