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Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R is defined to be the set of all prime ideals of R. It is denoted by Spec(R).

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

General schemes are obtained by "glueing together" several affine schemes.

Functoriality

Note that every ring homomorphism f : RS 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 (see category theory). Moreover for every prime P the homomorphism f descends to homomorphisms

O f-1P 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 fact can be used to define the functor Spec up to natural isomorphism.

Motivation from algebraic geometry

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 AK. 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 therefore 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. 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 eventually to schemes.

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