In algebra, the length of a module over a ring
is a generalization of the dimension of a vector space which measures its size.
[1] page 153 It is defined to be the length of the longest chain of submodules.
For vector spaces (modules over a field), the length equals the dimension.
can have finite length only when the module has Krull dimension zero.
Modules of finite length are called 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.
considered as a module over itself by left multiplication.
is the length of the longest chain of prime ideals.
Since all Artinian rings are Noetherian, this implies that a ring has finite length if and only if it is Artinian.
In particular, it implies the following two properties 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. Any finite dimensional vector space
In fact, these examples serve as the basic tools for defining the order of vanishing in intersection theory.
(viewed as a module over the integers Z) is equal to the number of prime factors of
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 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.
A special case of this general definition of a multiplicity is the order of vanishing of a non-zero algebraic function
of codimension 1[3] the order of vanishing for a polynomial
is the local ring defined by the stalk of
This idea can then be extended to rational functions
which is similar to defining the order of zeros and poles in complex analysis.
For example, consider a projective surface
, then the order of vanishing of a rational function
is a unit, so the quotient module is isomorphic to
This can be found using the maximal proper sequence
The order of vanishing is a generalization of the order of zeros and poles for meromorphic functions in complex analysis.
This kind of information can be encoded using the length of modules.
is a unit, so this is isomorphic to the quotient module
which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.