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.
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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 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 D_{f} be the set of all prime ideals P in Spec(R) such that f is not in P. The sets {D_{f}}_{f∈R} form a basis for the topology on Spec(R). Define a sheaf O_{X} on the D_{f} by setting Γ(D_{f}, O_{X}) = R_{f}, the localization of R at the multiplicative system {1,f,f^{2},f^{3},...}. It can be shown that this satisfies the necessary axioms to be a BSheaf. Next, if U is the union of {D_{fi}}_{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.
If R is an integral domain, with field of fractions K, then we can describe the ring Γ(U,O_{X}) 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,O_{X}) as precisely the set of elements of K which are regular at every point P in U.
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