Spectrum of a ring

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In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is 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.

Contents

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 an ideal 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).

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. However, Spec(R) is always a Kolmogorov space. It is also a spectral space.

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}fR form a basis for the topology on Spec(R). Define a sheaf OX on the Df by setting Γ(Df, OX) = Rf, 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 {Dfi}iI, we let Γ(U,OX) = limiI Rfi, and this produces a sheaf; see the sheaf article for more detail.

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.

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