Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with a topology, the Zariski topology, and with a structure sheaf, turning it into a locally ringed space.

Table of contents

1 Zariski topology 2 Sheaves and schemes 3 Functoriality 4 Example 5 Motivation from algebraic geometry

Zariski topology

Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R such that V consists of all those prime ideals in R that contain I. This is called the Zariski topology on Spec(R).

Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. Spec(R) is always a T₀ space, however.

Sheaves and schemes

To every open set U of Spec(R), one may assign a commutative ring R_U in the following way: let S be the complement of the union of all the prime ideals in U. Then S is a multiplicative set and we define R_U as the ring of quotients of R with respect to S. This endows Spec(R) with a sheaf of rings O. If P is an element of Spec(R), then the stalk O_P at P of this sheaf is equal to the localization of R at P, which is a local ring. Thus Spec(R) is a locally ringed space.

Every sheaf of rings of this form is called an affine scheme. General schemes are obtained by "gluing together" several affine schemes.

Functoriality

It is useful to use the language of category theory and observe that Spec is a functor. Every ring homomorphism f : R → S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a prime ideal in R). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover for every prime P the homomorphism f descends to homomorphisms

O_f ^-1(P) → O_P,

of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In fact it is the universal such functor and this can be used to define the functor Spec up to natural isomorphism.

The functor Spec yields a contravariant equivalence between the category of commutative rings and the category of affine schemes.

Example

A special but quite typical case of an affine scheme is obtained as follows. Take a field K and n variables, x₁,...,x_n. Given m polynomials, p₁,...,p_m in these variables over K, there is a functor F from the category of commutative K-algebras to sets characterized by F(A)={(x₁,...,x_n) in Aⁿ|p₁=...=p_m=0}. Then F is represented by Spec(B) where B is the quotient of K[x₁,...,x_n] by the ideal I generated by the p_j.

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kⁿ (where K is an algebraically closed field) which are defined as the common zeros of a set of polynomials in n variables. If A is such an algebraic set, one considers the commutative ring R of all polynomial functions A → K. The maximal ideals of R correspond to the points of A (because K is algebraically closed), and the prime ideals or R correspond to the subvarieties of A (an algebraic set is called irreducible or a variety if it cannot be written as the union of two proper algebraic subsets).

The spectrum of R therefore consists of the points of A together with elements for all subvarieties of A. The points of A are closed in the spectrum, while the elements corresponding to subvarieties have a closure consisting of all their points and subvarieties. If one only considers the points of A, i.e. the maximal ideals in R, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets (which has precisely the algebraic subsets as closed sets).

One can thus view the topological space Spec(R) as an "enrichment" of the topological space A (with Zariski topology): for every subvariety of A, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding subvariety. One thinks of this point as the "generic point" for the subvariety. Furthermore, the sheaf on Spec(R) and the sheaf of polynomial functions on A are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with Zariski topology, one can generalize the concepts of algebraic geometry to non-algebraically closed fields and beyond, eventually arriving at the language of schemess.

External links:

• Kevin R. Coombes: The Spectrum of a Ring, http://odin.mdacc.tmc.edu/~krc/agathos/spec.html