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. 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.
Definition.
Length of a module.
Let formula_1 be a (left or right) module over some ring formula_2. Given a chain of submodules of formula_1 of the form
we say that formula_5 is the length of the chain. The length of formula_1 is defined to be the largest length of any of its chains. If no such largest length exists, we say that formula_1 has infinite length.
Length of a ring.
A ring formula_2 is said to have finite length as a ring if it has finite length as a left formula_2-module.
Properties.
Finite length and finite modules.
If an formula_2-module formula_1 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.
An formula_2-module formula_1 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.
Supposeformula_14is a short exact sequence of formula_2-modules. Then M has finite length if and only if "L" and "N" have finite length, and we have formula_16 In particular, it implies the following two properties
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.
Any finite dimensional vector space formula_19 over a field formula_20 has a finite length. Given a basis formula_21 there is the chainformula_22which is of length formula_5. It is maximal because given any chain,formula_24the dimension of each inclusion will increase by at least formula_25. Therefore, its length and dimension coincide.
Artinian modules.
Over a base ring formula_2, 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 formula_27 (viewed as a module over the integers Z) is equal to the number of prime factors of formula_5, with multiple prime factors counted multiple times. This can be found by using the Chinese remainder theorem.
Use in multiplicity theory.
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 algebraic hypersurfaces 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.
A special case of this general definition of a multiplicity is the order of vanishing of a non-zero algebraic function formula_29 on an algebraic variety. Given an algebraic variety formula_30 and a subvariety formula_19 of codimension 1 the order of vanishing for a polynomial formula_32 is defined asformula_33where formula_34 is the local ring defined by the stalk of formula_35 along the subvariety formula_19 pages 426-227, or, equivalently, the stalk of formula_35 at the generic point of formula_19 page 22. If formula_30 is an affine variety, and formula_19 is defined the by vanishing locus formula_41, then there is the isomorphismformula_42This idea can then be extended to rational functions formula_43 on the variety formula_30 where the order is defined asformula_45which is similar to defining the order of zeros and poles in Complex analysis.
Example on a projective variety.
For example, consider a projective surface formula_46 defined by a polynomial formula_47, then the order of vanishing of a rational functionformula_48is given byformula_49whereformula_50For example, if formula_51 and formula_52 and formula_53 thenformula_54since formula_55 is a Unit (ring theory) in the local ring formula_56. In the other case, formula_57 is a unit, so the quotient module is isomorphic toformula_58so it has length formula_59. This can be found using the maximal proper sequenceformula_60
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 analysis. For example, the functionformula_61has zeros of order 2 and 1 at formula_62 and a pole of order formula_25 at formula_64. This kind of information can be encoded using the length of modules. For example, setting formula_65 and formula_66, there is the associated local ring formula_34 is formula_68 and the quotient module formula_69Note that formula_70 is a unit, so this is isomorphic to the quotient moduleformula_71Its length is formula_59 since there is the maximal chainformula_73of submodules. More generally, using the Weierstrass factorization theorem a meromorphic function factors asformula_48which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.