Length of a module

In algebra, the length of a module over a ring 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. For vector spaces, the length equals the dimension. If is an algebra over a field, the length of a module is at most its dimension as a -vector space.
In commutative algebra and algebraic geometry, a module over a Noetherian commutative ring 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

Let be a module over some ring. Given a chain of submodules of of the form
one says that is the length of the chain. The length of is the largest length of any of its chains. If no such largest length exists, we say that has infinite length. Clearly, if the length of a chain equals the length of the module, one has and

Length of a ring

The length of a ring is the length of the longest chain of ideals; that is, the length of considered as a module over itself by left multiplication. By contrast, the Krull dimension of is the length of the longest chain of [Prime ideal|prime ideals].

Properties

Finite length and finite modules

If an -module 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 -module has finite length if and only if it is both a Noetherian module and an Artinian module. 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

Supposeis 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 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 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 over a field has a finite length. Given a basis there is the chainwhich 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 modules

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.

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 is equal to the number of prime factors of, with multiple prime factors counted multiple times. This follows from the fact that the submodules of are in one to one correspondence with the positive divisors of, this correspondence resulting itself from the fact that 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 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 on an algebraic variety. Given an algebraic variety and a subvariety of codimension 1 the order of vanishing for a polynomial is defined aswhere is the local ring defined by the stalk of along the subvariety ^{pages 426-227}, or, equivalently, the stalk of at the generic point of ^{page 22}. If is an affine variety, and is defined the by vanishing locus, then there is the isomorphismThis idea can then be extended to rational functions on the variety 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 defined by a polynomial, then the order of vanishing of a rational functionis given bywhereFor example, if and and thensince is a unit in the local ring. In the other case, is a unit, so the quotient module is isomorphic toso it has length. This can be found using the maximal proper sequence

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 functionhas 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 moduleIts length is since there is the maximal chainof submodules. More generally, using the Weierstrass factorization theorem a meromorphic function factors aswhich is a product of linear polynomials in both the numerator and denominator.