In algebra, the length of a module over a ring <math>R</math> is a generalization of the dimension of a vector space which measures its size. <sup>page 153</sup> 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. If <math>R</math> is an algebra over a field <math>k</math>, the length of a module is at most its dimension as a <math>k</math>-vector space.

In commutative algebra and algebraic geometry, a module over a Noetherian commutative ring <math>R</math> 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 <math>M</math> be a (left or right) module over some ring <math>R</math>. Given a chain of submodules of <math>M</math> of the form

:<math>M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n,</math>

one says that <math>n</math> is the length of the chain. If R is a field, then the converse is also true.

Relation to Artinian and Noetherian modules

An <math>R</math>-module <math>M</math> has finite length if and only if it is both a Noetherian module and an Artinian module

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 <math>\mathbb{Z}/n\mathbb{Z}</math> (viewed as a module over the integers Z) is equal to the number of prime factors of <math>n</math>, with multiple prime factors counted multiple times. This follows from the fact that the submodules of <math>\mathbb{Z}/n\mathbb{Z}</math> are in one to one correspondence with the positive divisors of <math>n</math>, this correspondence resulting itself from the fact that <math>\Z</math> 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 <math>f \in R(X)^*</math> on an algebraic variety. Given an algebraic variety <math>X</math> and a subvariety <math>V</math> of codimension 1<math display=block>\operatorname{ord}_V(f) = \text{length}_{\mathcal{O}_{V,X\left( \frac{\mathcal{O}_{V,X{(f)} \right)</math>where <math>\mathcal{O}_{V,X}</math> is the local ring defined by the stalk of <math>\mathcal{O}_X</math> along the subvariety <math>V</math> <sup>page 22</sup>. If <math>X</math> is an affine variety, and <math>V</math> is defined the by vanishing locus <math>V(f)</math>, then there is the isomorphism<math display=block>\mathcal{O}_{V,X} \cong R(X)_{(f)}</math>This idea can then be extended to rational functions <math>F = f/g</math> on the variety <math>X</math> where the order is defined as More generally, using the Weierstrass factorization theorem a meromorphic function factors as<math display=block>F = \frac{f}{g}</math>which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.

See also

  • Hilbert–Poincaré series
  • Weil divisor
  • Chow ring
  • Intersection theory
  • Weierstrass factorization theorem
  • Serre's multiplicity conjectures
  • Hilbert scheme - can be used to study modules on a scheme with a fixed length
  • Krull–Schmidt theorem

References

  • Steven H. Weintraub, Representation Theory of Finite Groups AMS (2003) ,
  • Allen Altman, Steven Kleiman, A term of commutative algebra.
  • The Stacks project. Length