Spectrum of a commutative unital ring

Definition

The spectrum of a commutative unital ring is a highly structured object that captures much of the geometry associated with the ring. We here describe its structure at various levels.

Set-theoretic structure

Set-theoretically, the spectrum is the set of prime ideals in the ring.

Topological structure

A subset in the spectrum is deemed a closed set if and only if there exists a radical ideal of the ring such that the given subset is precisely the set of primes of the ring. Since every radical ideal is the intersection of the prime ideals containing it, there is a bijective correspondence between closed subsets of the spectrum and radical ideals of the ring.

This topology is termed the Zariski topology. The Zariski topology for the polynomial ring over an algebraically closed field, coincides with its Zariski topology in the other sense.

Structure of a locally ringed space

The spectrum can also be given the structure of a locally ringed space. The structure is as follows:

• To each open set, associate the localization at the multiplicative system which is the complement of the union of prime ideals corresponding to points in the open set.
• The restriction maps of the sheaf is the natural map from the localization at a smaller multiplicative system, to the localization at a larger multiplicative system
• The stalk at each point is a local ring, which is the localization at the prime ideal represented by that point

A locally ringed space that can be realized as the spectrum of a commutative unital ring in the above fashion, is termed an affine scheme.

Related notions

• Max-spectrum of a commutative unital ring

Facts

Topology of spectrum captures only the reduced part

The topological space structure of the spectrum ignores nilpotents, in the sense, that it depends only on the quotient of the ring by its nilradical (viz, the corresponding reduced ring).

Topological space properties of the spectrum

The spectrum satisfies the following topological properties:

• It is a $T_{0}$ space: This is direct
• It is a compact space: This follows essentially from the fact that every proper ideal is contained in a maximal ideal
• It is a sober space: This follows essentially from the fact that any irreducible radical ideal is prime

Correspondence between ideal properties and topological properties of subsets

Under the bijective correspondence between radical ideals and closed subsets, the following property correspondence is established:

• Prime ideals correspond to the closure of one-point subsets
• Maximal ideals correspond to closed points
• Minimal prime ideals correspond to those closuers of one-point subsets that are not contained in the closures of any other one-point subset

Correspondence between ring properties and properties of the spectrum

The converse in each case holds if we further assume that the ring is a reduced ring:

• The spectrum of a Noetherian ring is a Noetherian space
• The spectrum of an integral domain is one where there is a dense point (the point corresponding to the zero ideal)
• The spectrum of a field is a single point
• The spectrum of a zero-dimensional ring is a $T_{1}$ space
• The spectrum of a Jacobson ring is one where every closed set is the closure of the subset of its closed points