Spectrum of a ring

The spectrum of a ring is a term used in algebra, a branch of mathematics. The spectrum of the ring is in the set of all primes in character. He referred to the appropriate ring geometric object.

This article deals with commutative algebra. In particular, all rings considered are commutative and have an identity element. For more details see Commutative Algebra.

Definition

For a ring spectrum is a topological space with a sheaf of rings:

  • The space the underlying set is the set of prime ideals of.
  • The topology is the Zariski topology in which a base of open sets of quantities
  • The sections of the structure sheaf on are the same localization. It is particularly

Local ringed spaces which are isomorphic to the spectrum of a ring are called affine schemes.

Examples

  • The spectrum of a body consisting of a single point; the sections of the structure sheaf on this point are the same as the body itself
  • Consists of the 0 and the (positive ) primes; open sets are complements of a finite set of primes; the sections of the structure sheaf on such an open set are the rational numbers whose denominators contain only prime factors of.
  • The -dimensional affine space over a ring is the affine scheme. Is an algebraically closed field, then correspond to the closed points ( equivalent: the maximal ideals ) bijectively to points in space ( See: Hilbert Nullstellensatz ).
  • Be a compact Hausdorff space and let the ring of complex-valued continuous functions, then correspond to the closed points in the spectrum bijective points in the Hausdorff space. You can embed this way the Hausdorff topological space in the (generally non- Hausdorff ) space. This example treated here the spectrum of the ring theory to the Gelfand spectrum of a Banach algebra connects as it is examined and used in the functional analysis and operator theory.

Properties

  • The spectrum of a ring is a locally ringed space: the stalk of the structure sheaf at a point is the local ring.
  • The spectrum of a ring is always quasi- compact.
  • The formation of the spectrum is a contravariant functor: For a ring homomorphism is continuous, more precisely a homomorphism locally ringed spaces.
  • The functor Spec is a category equivalence between the category of rings ( commutative with unity) and the category of affine schemes, in particular, every morphism of affine schemes of the form of a ring homomorphism.
Robin Hartshorne
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