Spectrum of a ring

A topological space whose points are the prime ideals of a ring with the Zariski topology (also called the spectral topology). It is assumed that is commutative and has an identity. The elements of can be regarded as functions on by setting . supports a sheaf of local rings , called its structure sheaf. For a point , the stalk of over is the localization of at .

To any identity-preserving ring homomorphism there corresponds a continuous mapping . If is the nil radical of , then the natural mapping is a homeomorphism of topological spaces.

For a non-nilpotent element , let , where . Then the ringed spaces and , where is the localization of with respect to , are isomorphic. The sets are called the principal open sets. They form a basis for the topology on . A point is closed if and only if is a maximal ideal of . By assigning to its closure in , one obtains a one-to-one correspondence between the points of and the set of closed irreducible subsets of . is quasi-compact, but usually not Hausdorff. The dimension of is defined as the largest for which there is a sequence of distinct closed irreducible sets .

Many properties of can be described in terms of . For example, is Noetherian if and only if has the descending chain condition for closed sets; is an irreducible space if and only if is an integral domain; the dimension of coincides with the Krull dimension of , etc.

Sometimes one considers the maximal spectrum , which is the subspace of consisting of the closed points. For a graded ring one also considers the projective spectrum . If , then the points of are the homogeneous prime ideals of such that .

References

[1]	N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , XXVIII , Hermann (1961)
[2]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian)

Comments

The continuous mapping defined by a unitary ring homomorphism is given by .

The pair is an affine scheme.

Similarly, supports a sheaf of local rings , the stalk of which at a point is the homogeneous localization of at . (See also Localization in a commutative algebra.) The pair is a projective scheme.

Spectra have also been studied for non-commutative rings, cf. [a1].

For Krull dimension see Dimension (of an associative ring).

References

[a1]	F. van Oystaeyen, A. Verschoren, "Non-commutative algebraic geometry" , Lect. notes in math. , 887 , Springer (1981)