In abstract algebra and algebraic geometry, the **spectrum** of a commutative ring *R*, denoted by
*R*. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A ringed space of this form is called an **affine scheme**.

## Contents

## Zariski topology

For any ideal *I* of *R*, define
*I*. We can put a topology on

This topology is called the Zariski topology.

A basis for the Zariski topology can be constructed as follows. For *f*∈*R*, define *D*_{f} to be the set of prime ideals of *R* not containing *f*. Then each *D*_{f} is an open subset of

*R* are precisely the closed points in this topology. By the same reasoning, it is not, in general, a T_{1} space. However,
_{0} axiom); it is also a spectral space.

## Sheaves and schemes

Given the space
**structure sheaf** *O*_{X} is defined on the distinguished open subsets *D*_{f} by setting Γ(*D*_{f}, *O*_{X}) = *R*_{f}, the localization of *R* by the powers of *f*. It can be shown that this defines a B-sheaf and therefore that it defines a sheaf. In more detail, the distinguished open subsets are a basis of the Zariski topology, so for an arbitrary open set *U*, written as the union of {*D*_{fi}}_{i∈I}, we set Γ(*U*,*O*_{X}) = lim_{i∈I} *R*_{fi}. One may check that this presheaf is a sheaf, so
**affine scheme**. General schemes are obtained by gluing affine schemes together.

Similarly, for a module *M* over the ring *R*, we may define a sheaf
*D*_{f},
*M*_{f}, using the localization of a module. As above, this construction extends to a presheaf on all open subsets of

If *P* is a point in
*P* equals the localization of *R* at the ideal *P*, and this is a local ring. Consequently,

If *R* is an integral domain, with field of fractions *K*, then we can describe the ring Γ(*U*,*O*_{X}) more concretely as follows. We say that an element *f* in *K* is regular at a point *P* in *X* if it can be represented as a fraction *f* = *a*/*b* with *b* not in *P*. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe Γ(*U*,*O*_{X}) as precisely the set of elements of *K* which are regular at every point *P* in *U*.

## Functorial Perspective

It is useful to use the language of category theory and observe that

of local rings. Thus

The functor
**category of commutative rings** and the **category of affine schemes**; each of these categories is often thought of as the opposite category of the other.

## Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies *algebraic sets*, i.e. subsets of *K*^{n} (where *K* is an algebraically closed field) that 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* of *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
*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
*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 schemes.

## Examples

## Global or relative Spec

There is a relative version of the functor
**Spec** of the sheaf.

Global Spec has a universal property similar to the universal property for ordinary Spec. More precisely, just as Spec and the global section functor are contravariant right adjoints between the category of commutative rings and schemes, global Spec and the direct image functor for the structure map are contravariant right adjoints between the category of commutative

where

## Example of Relative Spec

Suppose we want to parameterize a family of affine hyperelliptic curves over some projective space, then relative spec gives the correct tools for this. Let

where the composition of the bottom arrows is

gives us the corresponding hyperelliptic curve over our point

## Representation theory perspective

From the perspective of representation theory, a prime ideal *I* corresponds to a module *R*/*I*, and the spectrum of a ring corresponds to irreducible cyclic representations of *R,* while more general subvarieties correspond to possibly reducible representations that need not be cyclic. Recall that abstractly, the representation theory of a group is the study of modules over its group algebra.

The connection to representation theory is clearer if one considers the polynomial ring
*I,* or equivalently a module
*R* (cyclic meaning generated by 1 element as an *R*-module; this generalizes 1-dimensional representations).

In the case that the field is algebraically closed (say, the complex numbers), every maximal ideal corresponds to a point in *n*-space, by the nullstellensatz (the maximal ideal generated by
*K*-linear maps
*n* numbers, or equivalently a covector

Thus, points in *n*-space, thought of as the max spec of
*R,* while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then correspond to *infinite*-dimensional representations.

## Functional analysis perspective

The term "spectrum" comes from the use in operator theory. Given a linear operator *T* on a finite-dimensional vector space *V*, one can consider the vector space with operator as a module over the polynomial ring in one variable *R*=*K*[*T*], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of *K*[*T*] (as a ring) equals the spectrum of *T* (as an operator).

Further, the geometric structure of the spectrum of the ring (equivalently, the algebraic structure of the module) captures the behavior of the spectrum of the operator, such as algebraic multiplicity and geometric multiplicity. For instance, for the 2×2 identity matrix has corresponding module:

the 2×2 zero matrix has module

showing geometric multiplicity 2 for the zero eigenvalue, while a non-trivial 2×2 nilpotent matrix has module

showing algebraic multiplicity 2 but geometric multiplicity 1.

In more detail:

*T*spans the space);

## Generalizations

The spectrum can be generalized from rings to C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra. Notably, for a Hausdorff space, the algebra of scalars (the bounded continuous functions on the space, being analogous to regular functions) are a *commutative* C*-algebra, with the space being recovered as a topological space from
*non*-commutative C*-algebras yields noncommutative topology.