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 of 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. The spectrum of a ring and the associated affine scheme are both denoted by 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. An endomorphism of a finite-dimensional complex vector space generates the subring 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 of The polynomial factors into prime factors of the form where the are the eigenvalues of This establishes a bijective correspondence between the eigenvalues of and the primes ideals 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 has only zero as an eigenvalue, and the minimal polynomial where , but the spectrum as a point set is the singleton , the prime ideal containing , independently of the nilpotent degree .
The modern notion of a spectrum thus includes a structure sheaf, which consists of a prescription of the functions that lie over each of the prime ideals. In the case of , the point set of the spectrum is the singleton but the nilpotent degree is encoded by the stalk of the structure sheaf at this point, .
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.
The modern construction of 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
As a set, the spectrum of a commutative ring is the set of the prime ideals of . It is made a topological space, with each prime ideal 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 , letThe set of all form the closed sets of the Zariski topology on One gets exactly the same closed sets if one restricts the definition to subsets that are ideals, since if , the ideal generated by . In fact, only radical ideals need to be considered, as for any ideal and its radical
Given a closed set , the ideal 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).
Among the open sets, that is the sets of the form some are especially important: those of the form so that is taken to be a principal ideal generated by some they are sometimes called the distinguished open sets or principal open sets. One has alwaysSince every open set is of the form 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 generators of the ideal.
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 (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 (alternative notation: ) 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 . Technically, this can be expressed aswhere 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 localizationwhere is the set of the integer powers of . One observes that so that for some positive integer and , and is invertible in . This allows the canonical homomorphism to be defined as the localization mapping
For the other open sets, is defined asand 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 , the stalk 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 captures the local, infinitesimal behavior of an element of 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 ( an algebraically closed field), and sections can be thought of as generalizations of regular functions; i.e., functions such that for every , there is an open subset containing so that for some for all Here, is the ring of polynomial functions on , known as its coordinate ring; if is the ideal of all polynomials that are zero on , then The rings of regular functions over open subsets of an affine variety, in fact, form a sheaf of -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.
Two fundamental facts from the theory of affine varieties inform what the ring of global sections of an affine scheme should be: 1) the ring of global regular functions of affine variety is just the coordinate ring 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 itself. By considering arbitrary commutative rings instead of only coordinate rings (i.e., reduced, finitely generated -algebras) and taking all prime ideals of a ring, not just maximal ones, these facts motivate defining the ring of global sections of affine scheme to simply be :
In the general case, the sections of a structure sheaf of an affine scheme cannot be regarded as true "functions". Nonetheless, it is useful to think of an element as an object that behaves analogously to a "function" on by defining its value at in the field of fractions as
For the special case of the polynomial ring with spectrum evaluation of at maximal ideal under this definition corresponds to polynomial evaluation at , , as one might expect.
This assignment of a value to 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 ring of regular functions on is the localization of at By analogy, elements of the form in can be regarded as allowed "functions" on motivating the assignment as the starting point for the abstract definition of the structure sheaf of affine schemes given in the previous subsection.