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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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property 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 It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on

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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. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. This is called the Zariski topology on Spec(R). In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic

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. In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals Spec(R) is always a Kolmogorov space, however. In Topology and related branches of Mathematics, the T0 spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class It is a spectral space. In Mathematics, a Topological space X with Topology Ω is said to be spectral if 1 X is Compact and T0

Sheaves and schemes

To define a structure sheaf on Spec(R), first let Df be the set of all prime ideals P in Spec(R) such that f is not in P. The sets {Df}f∈R form a basis for the topology on Spec(R). Define a sheaf on the Df by setting Γ(Df, OX) = Rf, the localization of R at the multiplicative system {1,f,f2,f3,. In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. . . }. It can be shown that this satisfies the necessary axioms to be a B-Sheaf. In Mathematics, the gluing axiom is introduced to define what a sheaf F on a Topological space X must satisfy given that it is a Next, if U is the union of {Dfi}i∈I, we let Γ(U,OX) = limi∈I Rfi, and this produces a sheaf; see the sheaf article for more detail. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,OX) 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. In Complex analysis, see Holomorphic function. In Mathematics, a regular function in the sense of Algebraic geometry Using this definition, we can describe Γ(U,OX) as precisely the set of elements of K which are regular at every point P in U.

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. 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 what is called Consequently, Spec(R) is a locally ringed space. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on

Every locally ringed space isomorphic to one 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. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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 Ring theory or 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 Intuitively this is a function 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

Of -1(P) → OP,

of local rings. Thus Spec even defines a contravariant functor from the category of commutative rings to the category of locally ringed spaces. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In fact it is the universal such functor and this can be used to define the functor Spec up to natural isomorphism.

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 Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

Motivation from algebraic geometry

Following on from the example, in algebraic geometry one studies algebraic sets, i. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with 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. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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). In Mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is

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. In Mathematics, in the fields of General topology and particularly of Algebraic geometry, a generic point P of a Topological space 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.

Global Spec

There is a relative version of the functor Spec called global Spec, or relative Spec, and denoted by Spec. For a scheme Y, and a quasi-coherent sheaf of OY-algebras A, there is a unique scheme X, called Spec A, and a morphism f \colon X \to Y such that for every open affine U \subseteq Y, there is an isomorphism induced by f: f^{-1}(U) \cong \mathrm{Spec} A(U), and such that for an inclusion of open affines U \subseteq V, the restriction map f^{-1}(U) \to f^{-1}(V) is the restriction map A(V) \to A(U).

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External links

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