Spectrum of a ring

related topics
{math, number, function}
{math, energy, light}

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.


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.

Full article ▸

related documents
Product topology
Mean value theorem
Theory of computation
Least common multiple
Automata theory
Isomorphism theorem
De Morgan's laws
Golden ratio base
Borel algebra
Differential topology
Tree (data structure)
Algebraic topology
Special linear group
Knight's tour
Monotonic function
Generalized Riemann hypothesis
Operator overloading
Cauchy-Riemann equations
Column space
Cotangent space
Monotone convergence theorem
Conjunctive normal form
Axiom of regularity
Generating trigonometric tables
Cartesian product