In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all proper prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

Zariski topology

Spec(R) can be turned into a topological space as follows: a subset V of Spec(R) is closed if and only if there exists a subset I of R such that V consists of all those prime ideals in R that contain I. This is called the Zariski topology on Spec(R). Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This article needs to be cleaned up to conform to a higher standard of quality. ...

Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. Spec(R) is always a Kolmogorov space, however. It is a spectral space. In mathematics, a compact space is a space that resembles a closed and bounded subset of Euclidean space Rn in that it is small in a certain sense and contains all its limit points. The modern general definition calls a topological space compact if every open cover of it has... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In topology and related branches of mathematics, the T0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces. ... In mathematics, a topological space X is said to be spectral if 1) X is compact and T0; 2) The set C(X) of all compact-open subsets of (X,Ω) is a sublattice of Ω and a base for the topology; 3) X is sober, that is any nonempty...

Sheaves and schemes

To define a structure sheaf on Spec(R), we first define D_f to be the set of all prime ideals P in Spec(R) such that f is not in P. {D_f}_f∈R is a basis for the topology of open sets. We define a sheaf on the D_f by setting Γ(D_f, O_X)=R_f, the localization of R at the multiplicative system {1,f,f²,f³,...}. It can be shown that this satisfies the necessary axioms to be a B-Sheaf. Next, if U is the union of {B_i}_i∈I, we let Γ(U,O_X)=lim_i∈I R_fi, and this produces a sheaf; see the sheaf article for more detail. In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...

To obtain a direct description of Γ(U, O_X) for any open set U in X, we notice that the limit above has the universal property that if T is any commutative ring and R_fi→T is any system of maps which agree when restricted to R, then there is a unique map Γ(U,O_X)→T through which the given maps factor. Since each f_i maps to a unit in T, if we let S be the multiplicative set generated by {f_i}_i∈I, then by the universal property of localization we get a unique map S^-1RT through which each R_fi→T factors. This is the same universal property that Γ(U,O_X) has, so Γ(U,O_X)=S^-1R. In mathematics, A set S is a multiplicative set if xy is in S whenever x and y both are in S. Given a ring R, a multiplicative subset S of R is a multiplicative set such that 1 is in S. ...

To obtain an even more direct description of Γ(U, O_X), let S' be the complement in R of all the prime ideals in U. S' is a multiplicative set, since it is the intersection of the multiplicative sets RP, where P is a prime ideal in U. Each f_i is in S' , so S⊆S' . For the other inclusion, choose a g in S' , and suppose that g is not in S. Then g is not a unit in S^-1R, so we may find a prime ideal P of R which contains g and does not meet S. P must lie in U, but then by the definition of S' , g is not in S' . Consequently, Γ(U, O_X)=S' ^-1R.

While this direct description may appear useful, most operations on sheaves can more easily be carried out on B-sheaves, and since a B-sheaf can always be extended to a sheaf in the setting of schemes, it is usually more useful to work over basic open sets.

If P is a point in Spec(R), that is, a prime ideal, then the stalk at P equals the localization of R at P, and this is a local ring. Consequently, Spec(R) is a locally ringed space. In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

Every sheaf of rings of this form is called an affine scheme. General schemes are obtained by "gluing together" several affine schemes. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

Functoriality

It is useful to use the language of category theory and observe that Spec is a functor. Every ring homomorphism f : R → S induces a continuous map Spec(f) : Spec(S) → Spec(R) (since the preimage of any prime ideal in S is a prime ideal in R). In this way, Spec can be seen as a contravariant functor from the category of commutative rings to the category of topological spaces. Moreover for every prime P the homomorphism f descends to homomorphisms Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ...

O_f ^-1(P) → O_P,

of local rings. Thus Spec 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 this can be used to define the functor Spec up to natural isomorphism. In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...

The functor Spec 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. In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i.e. subsets of Kⁿ (where K is an algebraically closed field) which 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). Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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 Spec(R) 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 Spec(R) 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. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...

External link

• Kevin R. Coombes: The Spectrum of a Ring