Spectrum of a ring

The spectrum of a ring is a construct from algebra , a branch of mathematics . The spectrum of a ring is the set of all prime ideals in , in signs . It designates the geometric object corresponding to the ring. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle \operatorname {Spec} (R)=\{P\mid P\subseteq R{\text{ mit }}P{\text{ Primideal}}\}}$

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

definition

For a ring , the spectrum is a topological space with a sheaf of rings: ${\displaystyle R}$ ${\displaystyle \operatorname {Spec} R}$

• The set underlying space is the set of prime ideals of . ${\displaystyle R}$
• The topology is the Zariski topology , in which the open sets are based on the sets

${\displaystyle D(f)=\{P\in \operatorname {Spec} R\mid f\notin P\}}$

for elements of is given. ${\displaystyle f}$ ${\displaystyle R}$

• The cuts of the structural grain over are equal to the localization . In particular is ${\displaystyle {\mathcal {O}}_{\operatorname {Spec} R}}$ ${\displaystyle D(f)}$ ${\displaystyle R_{f}}$

${\displaystyle \Gamma (\operatorname {Spec} R,{\mathcal {O}}_{\operatorname {Spec} R})=R}$.

Locally small spaces that are isomorphic to the spectrum of a ring are called affine schemes .

Examples

• The spectrum of a body consists of a single point; the cuts of the structural grain above this point are equal to the body itself.
• ${\displaystyle \operatorname {Spec} \mathbb {Z} }$consists of the 0 and the (positive) prime numbers ; open sets are complements of a finite set of prime numbers ; the intersections of the structural grain over such an open set are the rational numbers whose denominators only contain prime factors from . ${\displaystyle S}$ ${\displaystyle S}$
• The -dimensional affine space over a ring is the affine scheme . If an algebraically closed body , then the closed points (equivalent: the maximum ideals ) correspond bijectively to the points in space (see: Hilbert's zero theorem ). ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle \operatorname {Spec} R[x_{1},\dotsc ,x_{n}]}$ ${\displaystyle R=k}$ ${\displaystyle k^{n}}$
• Let be a compact Hausdorff space and let the ring of complex-valued continuous functions be on , then the closed points in the spectrum correspond bijectively to the points in . In this way, the Hausdorff space can be topologically embedded in the (generally non-Hausdorffian) space . This example combines the spectrum of ring theory dealt with here with the Gelfand spectrum of a Banach algebra , as it is examined and used in functional analysis and operator theory. ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle \operatorname {Spec} R}$

properties

• The spectrum of a ring is a locally reduced space : the stalk of the structural grain at one point is the local ring . ${\displaystyle {\mathcal {O}}_{\operatorname {Spec} R}}$ ${\displaystyle P}$ ${\displaystyle R_{P}}$
• 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 of locally small spaces. ${\displaystyle \varphi \colon A\to B}$ ${\displaystyle \operatorname {Spec} (\varphi )\colon \operatorname {Spec} (B)\to \operatorname {Spec} (A)}$
• The functor Spec is a category equivalence between the category of rings (commutative with one) and the category of affine schemes; in particular, each morphism of affine schemes is of the form for a ring homomorphism . ${\displaystyle \operatorname {Spec} (\varphi )}$ ${\displaystyle \varphi }$

See also

• Spectrum (operator theory)

literature

• Robin Hartshorne : Algebraic Geometry . Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9