Spectrum of a ring

In algebra and algebraic geometry, the spectrum of a commutative ring "R", denoted by formula_1, is the set of all prime ideals of "R". It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.
Zariski topology.
For any ideal "I" of "R", define formula_2 to be the set of prime ideals containing "I". We can put a topology on formula_1 by defining the collection of closed sets to be
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 formula_1, and formula_6 is a basis for the Zariski topology.
formula_1 is a compact space, but almost never Hausdorff: in fact, the maximal ideals in "R" are precisely the closed points in this topology. By the same reasoning, it is not, in general, a T1 space. However, formula_1 is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.
Sheaves and schemes.
Given the space formula_9 with the Zariski topology, the 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 formula_1 is a ringed space. Any ringed space isomorphic to one of this form is called an 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 formula_11 on formula_1. On the distinguished open subsets set Γ("D""f", formula_11) = "M""f", using the localization of a module. As above, this construction extends to a presheaf on all open subsets of formula_1 and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf.
If "P" is a point in formula_1, that is, a prime ideal, then the stalk of the structure sheaf at "P" equals the localization of "R" at the ideal "P", and this is a local ring. Consequently, formula_1 is a locally ringed space.
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 formula_17 is a functor. Every ring homomorphism formula_18 induces a continuous map formula_19 (since the preimage of any prime ideal in formula_20 is a prime ideal in formula_21). In this way, formula_17 can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime formula_23 the homomorphism formula_24 descends to homomorphisms
of local rings. Thus formula_17 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 hence can be used to define the functor formula_17 up to natural isomorphism.
The functor formula_17 yields a contravariant equivalence between the 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). Specifically, the maximal ideals in "R", i.e. formula_29, together with the Zariski topology, is homeomorphic to "A" also with the Zariski topology.
One can thus view the topological space formula_1 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 formula_1 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 schemes.
Non-affine examples.
Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together. 
Non-Zariski topologies on a prime spectrum.
Some authors (notably M. Hochster) consider topologies on prime spectra other than Zariski topology.
First, there is the notion of constructible topology: given a ring "A", the subsets of formula_60 of the form formula_61 satisfy the axioms for closed sets in a topological space. This topology on formula_60 is called the constructible topology.
In , Hochster considers what he calls the patch topology on a prime spectrum. By definition, the patch topology is the smallest topology in which the sets of the forms formula_63 and formula_64 are closed.
Global or relative Spec.
There is a relative version of the functor formula_17 called global formula_17, or relative formula_17. If formula_20 is a scheme, then relative formula_17 is denoted by formula_70 or formula_71. If formula_20 is clear from the context, then relative Spec may be denoted by formula_73 or formula_74. For a scheme formula_20 and a quasi-coherent sheaf of formula_76-algebras formula_77, there is a scheme formula_78 and a morphism formula_79 such that for every open affine formula_80, there is an isomorphism formula_81, and such that for open affines formula_82, the inclusion formula_83 is induced by the restriction map formula_84. That is, as ring homomorphisms induce opposite maps of spectra, the restriction maps of a sheaf of algebras induce the inclusion maps of the spectra that make up the 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 formula_76-algebras and schemes over formula_20. In formulas,
where formula_88 is a morphism of schemes.
Example of a relative Spec.
The relative spec is the correct tool for parameterizing the family of lines through the origin of formula_89 over formula_90 Consider the sheaf of algebras formula_91 and let formula_92 be a sheaf of ideals of formula_93 Then the relative spec formula_94 parameterizes the desired family. In fact, the fiber over formula_95 is the line through the origin of formula_96 containing the point formula_97 Assuming formula_98 the fiber can be computed by looking at the composition of pullback diagrams
where the composition of the bottom arrows
gives the line containing the point formula_101 and the origin. This example can be generalized to parameterize the family of lines through the origin of formula_102 over formula_103 by letting formula_104 and formula_105
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 formula_106 or, without a basis, formula_107 As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of formula_108 corresponds to choosing a basis for the vector space. Then an ideal "I," or equivalently a module formula_109 is a cyclic representation of "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 formula_110 corresponds to the point formula_111). These representations of formula_112 are then parametrized by the dual space formula_113 the covector being given by sending each formula_108 to the corresponding formula_115. Thus a representation of formula_116 ("K"-linear maps formula_117) is given by a set of "n" numbers, or equivalently a covector formula_118
Thus, points in "n"-space, thought of as the max spec of formula_119 correspond precisely to 1-dimensional representations 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:
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) is a "commutative" C*-algebra, with the space being recovered as a topological space from formula_123 of the algebra of scalars, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any commutative C*-algebra can be realized as the algebra of scalars of a Hausdorff space in this way, yielding the same correspondence as between a ring and its spectrum. Generalizing to "non"-commutative C*-algebras yields noncommutative topology.