Spectrum of a ring on arvau

Wikipedia / Spectrum of a ring

In mathematics, and more specifically in commutative algebra and algebraic geometry, the prime spectrum (or simply the spectrum) of a commutative ring is the set of all prime ideals equipped with a topology called the Zariski topology. The spectrum of a commutative ring is naturally endowed with a sheaf of commutative rings, called the structure sheaf, which makes it a ringed space; that is, commutative rings are associated to every point and every open set, which satisfy some compatibility conditions. The structure formed by the spectrum of a commutative ring and the associated ringed space is called an affine scheme or .

Affine schemes are a basic tool of modern algebraic geometry, and specifically scheme theory. Indeed, schemes are built by "gluing together" affine schemes in a way that is very similar to the construction of manifolds by gluing together open subsets of a Euclidean space equipped with the ring of the continuous functions over them. The adjective "affine" in the phrase "affine scheme" comes from the fact that an affine algebraic variety can be identified with the affine scheme built over its ring of regular functions.

For the related, more general notion of prime and maximal spectra of lattices, see .

Historical motivation

The idea of the spectrum of a ring was introduced under that name by Alexander Grothendieck. It brought together several parallel historical threads. One, which motivates the use of the word "spectrum", comes from linear algebra and more general functional analysis, in which the spectrum is used to denote the eigenvalues of a linear transformation. The other came from commutative algebra, in which a topological space (the Zariski topology) had been in use to describe the structure of prime ideals. The final synthesis was the addition of a structure sheaf, which encodes local geometric information near each prime ideal.

The first motivation comes from linear algebra. A endomorphism of a finite-dimensional complex vector space of the ring of the endomorphisms of . This ring is commutative, and there is a surjective canonical ring homomorphism that maps to , where is the univariate polynomial ring. The kernel of this homomorphism is the principal ideal generated by the minimal polynomial are the eigenvalues This establishes a bijective correspondence between the eigenvalues of This allows identifying the spectrum of an endomorphism with the spectrum of a specific ring.

One missing ingredient in the notion of spectrum is that it does not distinguish the multiplicity of the various eigenvalues. Thus the definition of a ring spectrum includes not only the bare set of prime ideals, but also structural information, carried by the ring , that allows for the determination of multiplicity (and other kinds of geometric properties). For example, a non-zero nilpotent operator.

This linear algebra notion of spectrum was already widely used in the broader subject of spectral theory. In a unital complex Banach algebra , the idea is extended by defining the spectrum of an element to be the set of complex numbers such that is not invertible. In the commutative case, Israel Gelfand's theory of normed rings and Banach algebras made the space of maximal ideals, or equivalently multiplicative linear functionals, into a central object of study.

The prime spectrum of a commutative ring also has a separate origin in commutative algebra and algebraic geometry. For an affine algebraic variety over an algebraically closed field, the Nullstellensatz identifies ordinary points with maximal ideals in its coordinate ring. More generally, irreducible subvarieties correspond to prime ideals. Passing from maximal ideals to all prime ideals therefore amounts to adjoining a generic point for each irreducible subvariety.

The work of Wolfgang Krull anticipated the use of geometric points to describe prime ideals. His 1928 paper on chains of prime ideals was part of the development of dimension theory in general rings. A related topological use of prime ideals appeared in Marshall Stone's work on Boolean algebras and distributive lattices: Stone introduced a topology on prime ideals of a distributive lattice, producing what is now called the spectrum of the lattice and leading to the notion of a spectral space.

as a locally ringed space was introduced systematically by Alexander Grothendieck in scheme theory. In this form, the spectrum is not only a topological space of prime ideals, but is equipped with a structure sheaf, allowing affine schemes to be glued to form general schemes.

Zariski topology

of a commutative ring is the set of the prime ideals of . It is made a topological space being a point in this space, by equipping it with the Zariski topology, the topology for which a closed set is the set of all prime ideals containing a given subset of . In other words, for every subset of The set of all One gets exactly the same closed sets if one restricts the definition to subsets that are idealss, since if , the ideal generated by and its radical

Given a closed set is a radical ideal such that . This establishes a one-to-one correspondence between closed sets and radical ideals. This corresponds, in algebraic geometry, to the correspondence between an algebraic set and the set all polynomial equations that are satisfied on it (see Hilbert's Nullstellensatz for details).

they are sometimes called the distinguished open sets or principal open sets. the principal open sets form a basis for the Zariski topology. It follows that there is generally no harm to consider only open sets of the form . The importance of the lies mainly in the fact that, when an ideal is not principal, the open set is not easy to define in terms of the generatorss of the ideal.

is a compact space, but almost never Hausdorff: In fact, the maximal ideals are precisely the closed points is not, in general, a T1 space. is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space.

Affine schemes

Construction of the structure sheaf

For every commutative ring , the topological space is naturally endowed with a sheaf of commutative rings, called its structural sheaf and commonly denoted . This makes a ringed space, called an affine scheme and also denoted .

A sheaf of rings over a topological space consists of a family of rings ) indexed by the open sets of . For each inclusion of open sets, there is a canonical ring homomorphism . These canonical homomorphisms must satisfy some compatibility conditions.

The first compatibility condition is that the canonical homomorphisms behave as expected with respect to composition of inclusions. This means that if is the category of the open sets with inclusions as morphisms, is a contravariant functor from to the category of rings.

The second compatibility condition is that can be uniquely recovered from the if is an open cover of where denotes an inverse limit.

In the case of an affine scheme , the ring is first defined for principal open sets of the form as the localization is the set of the integer powers of and , and is invertible in . This allows the canonical homomorphism to be defined as the localization

For the other open sets, and the canonical ring homomorphisms are defined accordingly. These definitions, for open sets that are possibly not principal open sets, are rarely used in practice, except for proving that these definitions define effectively a sheaf of rings.

For a ringed space , the stalk at a point is the direct limitwhere runs over the open sets of containing . In the case of an affine scheme at a point is the local ring . Therefore, an affine scheme is a locally ringed space. The elements of a stalk are called germs. A germ at around .

Sections of the structure sheaf

The elements of are known as sections of the structure sheaf over . The ring of sections for an affine scheme can be thought of as a generalization of the ring of regular functions over open subset of an affine variety such that for every containing is the ring of polynomial functions on , known as its coordinate ring is the ideal of all polynomials that are zero on -algebras, which is called the structure sheaf of the variety. The structure sheaf of affine schemes generalizes this construction; for details, see Affine variety § Structure sheaf.

of affine variety 2) the Nullstellensatz establishes a one-to-one correspondence between the points of an affine variety and the maximal ideals of its coordinate ring . Thus, the coordinate ring gives us all of the allowed global functions on , while its set of maximal ideals could be identified with -algebras) and taking all prime to simply be

as an object that behaves analogously to a "function" on in the field of fractions as

under this definition corresponds to polynomial evaluation at , , as one might expect.

allows us to rewrite the definition of a principal open subset of as

which is now evidently analogous to the definition of a principal open subset of an affine variety the localization of as the starting point for the abstract definition of the structure sheaf of affine schemes given in the previous subsection.

Moreover, a nonzero more generally, ) cannot be interpreted as true "functions", except in special cases. Instead, elements of will be a collection of locally compatible. Here, locally compatible Formally, put

is an integral domain, with field of fractions then we can describe the ring Note that this agrees with the notion of a regular function in algebraic geometry. Using this definition, we can describe or, more explicitly,

This is equivalent to describing embedded in .

Category equivalence between rings and affine schemes

Affine schemes form a category whose morphisms are morphisms of ringed spaces. In this context, is a contravariant functor from the category of commutative rings to that of affine schemes. Conversely, given an affine scheme, the defining ring may be recovered as . This defines a functor in the opposite direction, and these two functors make the two categories dually equivalent.

Morphisms of affine schemes

A of ringed spaces from is formed by a continuous map from to , and, for every open subset of , a ring homomorphism ; moreover, these homomorphisms must commute with the homomorphisms defined by inclusions of open sets.

For making a functor, one must define it on ring homomorphisms.

So, let be a homomorphism of commutative rings and denote respectively by and the associated topological spaces. Since the points of these spaces are prime ideals, one may define a map by for every prime ideal of , since the inverse image of a prime ideal by a ring homomorphism is always a prime ideal. This map is continuous: for proving this one must prove that the inverse image of a closed subset of (where is any ideal of ) is the closed set (this results immediately from the monotonicity is homeomorphic to the principal open set to be .

To have a morphism of ringed spaces, one must define for each open subset of a ring homomorphism . In fact, it suffices to define this homomorphism on principal open sets, when . In this case, this homomorphism is the canonical ring homomorphism . It is straightforward, although rather lengthy, to verify all the compatibility conditions required for a morphism of ringed spaces.

Affine algebraic varieties

Affine algebraic varieties give foundational examples of affine schemes in the sense that Alexandre Grothendieck introduced scheme theory to provide a setting to clearly and precisely reformulate and resolve problems that were badly or inelegantly handled by the classical theory of varieties. Some of the issues that were addressed include dealing with multiplicities, developing a coordinate-free approach, studying the rational points over a field that is not algebraically closed, and establishing a single framework for affine, projective and abstract algebraic varieties.

An affine algebraic set over the field of complex numbers is the set of the common zeros in of a set of polynomials in indeterminates, that is, polynomials in . The set of the common zeros remains the same if the polynomials are replaced with the ideal they generate. The quotient ring , called the on , is isomorphic to the ring of the polynomial functions with values in , defined up to equality on . Indeed, Hilbert's Nullstellensatz establishes a homeomorphism for Zariski topologies between the points of and the maximal ideals of , that is, the closed points of . So, the points of may be identified with the maximal ideals of , and a regular function consists of the evaluation of an element of on the closed points of the spectrum.

In short, every statement about affine algebraic sets and affine algebraic varieties may be translated in the language of affine schemes. This has many advantages; in particular:

There is no need to suppose the base field is algebraically closed, and even to suppose the existence of a ground field (when dealing with , there is no need to suppose that contains a field.
There is no need to suppose that is an integral domain, as it usually the case in classical algebraic geometry. In particular, the intersection of two affine varieties is, in general, not a variety; it is an algebraic set with multiplicities. For example, the intersection of the circle and the line is , and the intersection is encoded in the affine scheme, while it is not in the set-theoretical definition of algebraic sets.

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets is an algebraically closed field) that are defined as the common zeros of a set of polynomials of all polynomial functions. The maximal ideals is algebraically closed), and the prime ideals correspond to the irreducible subvarieties (an algebraic set is called irreducible if it cannot be written as the union of two proper algebraic subsets).

, together with the Zariski topology, is homeomorphic also with the Zariski topology.

, 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 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

is the final object is the initial object for prime numbers
have been illustrated in a famous sketch by David Mumford, now known as 'Mumford's treasure map', found in his introductory textbook The Red Book of Varieties and Schemes.
. From the functor of points. This fundamental observation allows us to give meaning to other affine schemes.
.
The prime spectrum of a Boolean ring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that is, a Stone space).
(M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) 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 n-space. This can be easily generalized to any base ring, see Proj construction.
Affine plane minus the origin..

Non-Zariski topologies on a prime spectrum

Some authors (notably M. Hochster) consider topologies on prime spectra other than the Zariski topology.

First, there is the notion of constructible topology: given a ring A is called the constructible topology.

In , Hochster considers what he calls the patch topology on a prime spectrum. are closed.

Global or relative Spec

and a quasi-coherent \mathcal{O}_S-algebras. 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.

. In formulas,

is a morphism of schemes.

Example of a relative Spec

the fiber can be computed by looking at the composition of pullback diagrams

where the composition of the bottom arrows

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 As the latter formulation makes clear, a polynomial ring is the monoid algebra over a vector space corresponds to choosing a basis for the vector space. Then an ideal I, 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 are then parametrized by the dual space (K) is given by a set of n

Thus, points in n 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:

the eigenvalues (with geometric multiplicity) of the operator correspond to the (reduced) points of the variety, with multiplicity;
the primary decomposition of the module corresponds to the unreduced points of the variety;
a diagonalizable (semisimple) operator corresponds to a reduced variety;
a cyclic module (one generator) corresponds to the operator having a cyclic vector (a vector whose orbit under T spans the space);
the last invariant factor of the module equals the minimal polynomial of the operator, and the product of the invariant factors equals the characteristic polynomial.

Similar concepts

The spectrum can also be considered for C*-algebras in operator theory, yielding the notion of the spectrum of a C*-algebra is a unital commutative, indeed functorially so; this is the content of the Banach–Stone theorem. Indeed, any unital commutative C*-algebra can be realized as the ring of continuous functions of a compact 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.