In algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 It is defined to be the length of the longest chain of submodules-vector space.

In commutative algebra and algebraic geometry, a module over a Noetherian can have finite length only when the module has Krull dimension zero. Modules of finite length are finitely generated modules, but most finitely generated modules have infinite length. Modules of finite length are Artinian modules and are fundamental to the theory of Artinian rings.

The degree of an algebraic variety inside an affine or projective space is the length of the coordinate ring of the zero-dimensional intersection of the variety with a generic linear subspace of complementary dimension. More generally, the intersection multiplicity of several varieties is defined as the length of the coordinate ring of the zero-dimensional intersection.

Definition

Length of a module

be a (left or right) module over some ring of the form

is the length of the chain. The length has infinite length

Length of a ring

is the length of the longest chain of ideals considered as a module over itself by left multiplication. By contrast, the Krull dimension is the length of the longest chain of prime ideals.

Properties

Finite length and finite modules

has finite length, then it is finitely generated. If R is a field, then the converse is also true.

Relation to Artinian and Noetherian modules

has finite length if and only if it is both a Noetherian module and an Artinian module (cf. Hopkins' theorem). Since all Artinian rings are Noetherian, this implies that a ring has finite length if and only if it is Artinian.

Behavior with respect to short exact sequences

is a short exact sequence-modules. Then M has finite length if and only if L and N In particular, it implies the following two properties

• The direct sum of two modules of finite length has finite length

• The submodule of a module with finite length has finite length, and its length is less than or equal to its parent module.

Jordan–Hölder theorem

A composition series of the module M is a chain of the form

such that

A module M has finite length if and only if it has a (finite) composition series, and the length of every such composition series is equal to the length of M.

Examples

Finite dimensional vector spaces

. Therefore, its length and dimension coincide.

Artinian modules

, Artinian modules form a class of examples of finite modules. In fact, these examples serve as the basic tools for defining the order of vanishing in intersection theory.

Zero module

The zero module is the only one with length 0.

Simple modules

Modules with length 1 are precisely the simple modules.

Artinian modules over Z

The length of the cyclic group (viewed as a module over the integers Z) is equal to the number of prime is a principal ideal ring.

Use in multiplicity theory

For the needs of intersection theory, Jean-Pierre Serre introduced a general notion of the multiplicity of a point, as the length of an Artinian local ring related to this point.

The first application was a complete definition of the intersection multiplicity, and, in particular, a statement of Bézout's theorem that asserts that the sum of the multiplicities of the intersection points of algebraic hypersurfacess in a -dimensional projective space is either infinite or is exactly the product of the degrees of the hypersurfaces.

This definition of multiplicity is quite general, and contains as special cases most of previous notions of algebraic multiplicity.

Order of vanishing of zeros and poles

on an algebraic variety. Given an algebraic variety and a subvariety of codimension 1 is defined as pages 426-227, or, equivalently, the stalk page 22 is an affine varietyThis idea can then be extended to rational functions where the order is defined as which is similar to defining the order of zeros and poles in complex analysis.

Example on a projective variety

For example, consider a projective surface is a unit in the local ring

Zero and poles of an analytic function

The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in complex analysisof submodules. More generally, using the Weierstrass factorization theoremwhich is a (possibly infinite) product of linear polynomials in both the numerator and denominator.

See also

• Hilbert–Poincaré series

• Weil divisor

• Chow ring

• Intersection theory

• Weierstrass factorization theorem

• Serre's multiplicity conjectures

• Hilbert scheme - can be used to study modules on a scheme with a fixed length

• Krull–Schmidt theorem

References

External links

• Steven H. Weintraub, Representation Theory of Finite Groups AMS (2003) ,

• Allen Altman, Steven Kleiman, A term of commutative algebra.

• The Stacks project.