Spectrum of a ring

In commutative algebra, the prime spectrum of a commutative ring is the set of all prime ideals of, and is usually denoted by ; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings.

Zariski topology

For any ideal of, define to be the set of prime ideals containing. We can put a topology on 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, define to be the set of prime ideals of not containing. Then each is an open subset of, and is a basis for the Zariski topology.
is a compact space, but almost never Hausdorff: In fact, the maximal ideals in are precisely the closed points in this topology. By the same reasoning, is not, in general, a T₁ space. However, is always a Kolmogorov space ; it is also a spectral space.

Sheaves and schemes

Given the space with the Zariski topology, the structure sheaf is defined on the distinguished open subsets by setting the localization of by the powers of. 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, written as the union of, we set where denotes the inverse limit with respect to the natural ring homomorphisms One may check that this presheaf is a sheaf, so 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 over the ring, we may define a sheaf on. On the distinguished open subsets set using the localization of a module. As above, this construction extends to a presheaf on all open subsets of and satisfies the gluing axiom. A sheaf of this form is called a quasicoherent sheaf.
If is a point in, that is, a prime ideal, then the stalk of the structure sheaf at equals the localization of at the ideal, which is generally denoted, and this is a local ring. Consequently, is a locally ringed space.
If is an integral domain, with field of fractions, then we can describe the ring more concretely as follows. We say that an element in is regular at a point in if it can be represented as a fraction with. Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe as precisely the set of elements of that are regular at every point in.

Functorial perspective

It is useful to use the language of category theory and observe that is a functor. Every ring homomorphism induces a continuous map . In this way, can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover, for every prime the homomorphism descends to homomorphisms
of local rings. Thus 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 hence can be used to define the functor up to natural isomorphism.
The functor 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 that are defined as the common zeros of a set of polynomials in variables. If is such an algebraic set, one considers the commutative ring of all polynomial functions. The maximal ideals of correspond to the points of , and the prime ideals of correspond to the irreducible subvarieties of .
The spectrum of therefore consists of the points of together with elements for all irreducible subvarieties of. The points of 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, i.e. the maximal ideals in, then the Zariski topology defined above coincides with the Zariski topology defined on algebraic sets. Specifically, the maximal ideals in, i.e., together with the Zariski topology, is homeomorphic to also with the Zariski topology.
One can thus view the topological space as an "enrichment" of the topological space : for every irreducible subvariety of, one additional non-closed point has been introduced, and this point "keeps track" of the corresponding irreducible subvariety. One thinks of this point as the generic point for the irreducible subvariety. Furthermore, the structure sheaf on and the sheaf of polynomial functions on are essentially identical. By studying spectra of polynomial rings instead of algebraic sets with the 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

The spectrum of integers: The affine scheme is the final object in the category of affine schemes since is the initial object in the category of commutative rings.
The scheme-theoretic analogue of : The affine scheme. From the functor of points perspective, a point can be identified with the evaluation morphism. This fundamental observation allows us to give meaning to other affine schemes.
The cross: looks topologically like the transverse intersection of two complex planes at a point, although typically this is depicted as a, since the only well defined morphisms to are the evaluation morphisms associated with the points.
The prime spectrum of a Boolean ring is a compact totally disconnected Hausdorff space.
A topological space is homeomorphic to the prime spectrum of a commutative ring if and only if it is compact, quasi-separated and sober.
Non-affine examples

Here are some examples of schemes that are not affine schemes. They are constructed from gluing affine schemes together.

The projective -space over a field. This can be easily generalized to any base ring, see Proj construction. The projective -space for is not affine as the ring of global sections of is.
Affine plane minus the origin. Inside are distinguished open affine subschemes. Their union is the affine plane with the origin taken out. The global sections of are pairs of polynomials on that restrict to the same polynomial on, which can be shown to be, the global sections of. is not affine as in.
Non-Zariski topologies on a prime spectrum

Some authors consider topologies on prime spectra other than the Zariski topology.
First, there is the notion of constructible topology: given a ring A, the subsets of of the form satisfy the axioms for closed sets in a topological space. This topology on 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 and are closed.

Global or relative Spec

There is a relative version of the functor called global, or relative. If is a scheme, then relative is denoted by or. If is clear from the context, then relative Spec may be denoted by or. For a scheme and a quasi-coherent sheaf of -algebras, there is a scheme and a morphism such that for every open affine, there is an isomorphism, and such that for open affines, the inclusion is induced by the restriction map. 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 -algebras and schemes over. In formulas,
where 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 over Consider the sheaf of algebras and let be a sheaf of ideals of Then the relative spec parameterizes the desired family. In fact, the fiber over is the line through the origin of containing the point Assuming 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 and the origin. This example can be generalized to parameterize the family of lines through the origin of over by letting and

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 or, without a basis, As the latter formulation makes clear, a polynomial ring is the monoid algebra over a vector space, and writing in terms of corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module is a cyclic representation of R.
In the case that the field is algebraically closed, every maximal ideal corresponds to a point in n-space, by the Nullstellensatz. These representations of are then parametrized by the dual space the covector being given by sending each to the corresponding. Thus a representation of is given by a set of n numbers, or equivalently a covector
Thus, points in n-space, thought of as the max spec of correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations. The non-maximal ideals then correspond to infinite-dimensional representations.