Carrier of a module

In commutative algebra, the carrier of a module is the set of all prime ideals , so that the module does not become a null module after localization according to such a prime ideal .

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

If a unitary module is over a commutative ring with one and a prime ideal, then the localization of the module is described according to the prime ideal . With the set of all prime ideals of (see spectrum of a ring ). ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle p}$ ${\displaystyle M_{p}}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle \operatorname {Spec} (R)}$ ${\displaystyle R}$

The carrier of is defined as: ${\displaystyle M}$

${\displaystyle \operatorname {Supp} (M):=\{p\in \operatorname {Spec} (R)\mid M_{p}\neq \langle 0\rangle \}}$

(after English support for "carrier")

sentences

Isolation of the wearer

The annihilator of is: ${\displaystyle M}$

${\displaystyle \operatorname {Ann} M=\{a\in A\mid am=0\ \mathrm {f{\ddot {u}}r\ alle} \ m\in M\}}$

The following sentence applies:

• If it is finitely generated, then: ${\displaystyle M}$

${\displaystyle \operatorname {Supp} (M)=\{p\in \operatorname {Spec} (R)\mid p\supset \operatorname {Ann} (M)\}}$

In particular, the carrier of in this case is a closed set of . ${\displaystyle M}$ ${\displaystyle \operatorname {Spec} (R)}$

Local-global principle

The carrier of a module that is not the zero module is not empty. The local-global statement applies that the following three statements are equivalent:

• The following applies to all maximum ideals : ${\displaystyle m}$

${\displaystyle M_{m}=0}$

• The following applies to all prime ideals : ${\displaystyle p}$

${\displaystyle M_{p}=0}$

• It is

${\displaystyle M=0}$

A module is therefore the null module if and only if it is locally the null module.

literature

• Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry. Vieweg, 1980, ISBN 3-528-07246-6 .
• Atiyah, Macdonald: Introduction to Commutative Algebra. Addison-Wesley, 1969, ISBN 0-2010-0361-9 .