In algebra, the length of a module over a ring is a generalization of the
submodules
. For vector spaces (modules over a field), 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.
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 (left or right) module over some ring. Given a chain of submodules of of the form
one says that is the length of the chain.[1] 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 ideals.
Properties
Finite length and finite modules
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 modules
An -module has finite length if and only if it is both a
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
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.
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 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.[3]
The length of the cyclic group (viewed as a module over the integersZ) 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
Main article:
Intersection multiplicity
For the needs of
Artinian local ring
related to this point.
The first application was a complete definition of the
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
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 isomorphismThis 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 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.[6] More generally, using the Weierstrass factorization theorem a meromorphic function factors aswhich is a (possibly infinite) product of linear polynomials in both the numerator and denominator.