Length of a module

In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size.[1] page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces.

Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory whereas length is used to analyze finite modules. There are also various ideas of dimension that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry and Deformation theory where Artin rings are used extensively.

DefinitionEdit

Length of a moduleEdit

Let   be a (left or right) module over some ring  . Given a chain of submodules of   of the form

 

we say that   is the length of the chain.[1] The length of   is defined to be the largest length of any of its chains. If no such largest length exists, we say that   has infinite length.

Length of a ringEdit

A ring   is said to have finite length as a ring if it has finite length as a left  -module.

PropertiesEdit

Finite length and finite modulesEdit

If an  -module   has finite length, then it is finitely generated.[2] If R is a field, then the converse is also true.

Relation to Artinian and Noetherian modulesEdit

An  -module   has finite length if and only if it is both a Noetherian module and an Artinian module[1] (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 sequencesEdit

Suppose

 

is a short exact sequence of  -modules. Then M has finite length if and only if L and N have finite length, and we have

 

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 theoremEdit

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.

ExamplesEdit

Finite dimensional vector spacesEdit

Any finite dimensional vector space   over a field   has a finite length. Given a basis   there is the chain

 

which is of length  . It is maximal because given any chain,

 

the dimension of each inclusion will increase by at least  . Therefore its length and dimension coincide.

Artinian modulesEdit

Over a base ring  , 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.[3]

Zero moduleEdit

The zero module is the only one with length 0.

Simple modulesEdit

Modules with length 1 are precisely the simple modules.

Artinian modules over ZEdit

The length of the cyclic group   (viewed as a module over the integers Z) is equal to the number of prime factors of  , with multiple prime factors counted multiple times. This can be found by using the Chinese remainder theorem.

Use in multiplicity theoryEdit

For the need 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 n algebraic hypersurfaces in a n-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 polesEdit

A special case of this general definition of a multiplicity is the order of vanishing of a non-zero algebraic function   on an algebraic variety. Given an algebraic variety   and a subvariety   of codimension 1[3] the order of vanishing for a polynomial   is defined as[4]

 

where   is the local ring defined by the stalk of   along the subvariety  [3] pages 426-227, or, equivalently, the stalk of   at the generic point of  [5] page 22. If   is an affine variety, and   is defined the by vanishing locus  , then there is the isomorphism

 

This idea can then be extended to rational functions   on the variety   where the order is defined as

 [3]

which is similar to defining the order of zeros and poles in Complex analysis.

Example on a projective varietyEdit

For example, consider a projective surface   defined by a polynomial  , then the order of vanishing of a rational function

 

is given by

 

where

 

For example, if   and   and   then

 

since   is a Unit (ring theory) in the local ring  . In the other case,   is a unit, so the quotient module is isomorphic to

 

so it has length  . This can be found using the maximal proper sequence

 

Zero and poles of an analytic functionEdit

The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in Complex analysis. For example, the function

 

has zeros of order 2 and 1 at   and a pole of order   at  . This kind of information can be encoded using the length of modules. For example, setting   and  , there is the associated local ring   is   and the quotient module

 

Note that   is a unit, so this is isomorphic to the quotient module

 

Its length is   since there is the maximal chain

 

of submodules.[6] More generally, using the Weierstrass factorization theorem a meromorphic function factors as

 

which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.

See alsoEdit

ReferencesEdit

  1. ^ a b c "A Term of Commutative Algebra". www.centerofmathematics.com. pp. 153–158. Archived from the original on 2013-03-02. Retrieved 2020-05-22. Alt URL
  2. ^ "Lemma 10.51.2 (02LZ)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-22.
  3. ^ a b c d Fulton, William, 1939- (1998). Intersection theory (2nd ed.). Berlin: Springer. pp. 8–10. ISBN 3-540-62046-X. OCLC 38048404.CS1 maint: multiple names: authors list (link)
  4. ^ "Section 31.26 (0BE0): Weil divisors—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-22.
  5. ^ Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. New York, NY: Springer New York. doi:10.1007/978-1-4757-3849-0. ISBN 978-1-4419-2807-8.
  6. ^ "Section 10.120 (02MB): Orders of vanishing—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-22.

External linksEdit