Spectrum of a ring

In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme. In algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme. For any ideal I of R, define V I {displaystyle V_{I}} to be the set of prime ideals containing I. We can put a topology on Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} 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 f∈R, define Df to be the set of prime ideals of R not containing f. Then each Df is an open subset of Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} , and { D f : f ∈ R } {displaystyle {D_{f}:fin R}} is a basis for the Zariski topology. Spec ⁡ ( R ) {displaystyle operatorname {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. By the same reasoning, it is not, in general, a T1 space. However, Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} is always a Kolmogorov space (satisfies the T0 axiom); it is also a spectral space. Given the space X = Spec ⁡ ( R ) {displaystyle X=operatorname {Spec} (R)} with the Zariski topology, the structure sheaf OX is defined on the distinguished open subsets Df by setting Γ(Df, OX) = Rf, the localization of R by the powers of f. 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 U, written as the union of {Dfi}i∈I, we set Γ(U,OX) = limi∈I Rfi. One may check that this presheaf is a sheaf, so Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} 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 M over the ring R, we may define a sheaf M ~ {displaystyle { ilde {M}}} on Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} . On the distinguished open subsets set Γ(Df, M ~ {displaystyle { ilde {M}}} ) = Mf, using the localization of a module. As above, this construction extends to a presheaf on all open subsets of Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} and satisfies gluing axioms. A sheaf of this form is called a quasicoherent sheaf. If P is a point in Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} , that is, a prime ideal, then the stalk of the structure sheaf at P equals the localization of R at the ideal P, and this is a local ring. Consequently, Spec ⁡ ( R ) {displaystyle operatorname {Spec} (R)} is a locally ringed 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. 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.