Length of a module

Let ${\displaystyle M}$ be a (left or right) module over some ring ${\displaystyle R}$. Given a chain of submodules of ${\displaystyle M}$ of the form

${\displaystyle M_{0}\subsetneq M_{1}\subsetneq \cdots \subsetneq M_{n},}$

one says that ${\displaystyle n}$ is the length of the chain.^[1] The length of ${\displaystyle M}$ is the largest length of any of its chains. If no such largest length exists, we say that ${\displaystyle M}$ has infinite length. Clearly, if the length of a chain equals the length of the module, one has ${\displaystyle M_{0}=0}$ and ${\displaystyle M_{n}=M.}$

Length of a ring

A ring ${\displaystyle R}$ is said to have finite length as a ring if it has finite length as a left ${\displaystyle R}$-module.

Finite length and finite modules

If an ${\displaystyle R}$-module ${\displaystyle M}$ 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 ${\displaystyle R}$-module ${\displaystyle M}$ has finite length if and only if it is both a Noetherian module and an Artinian module^[1] (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

Suppose

$${\displaystyle 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0}$$

is a short exact sequence of ${\displaystyle R}$-modules. Then M has finite length if and only if L and N have finite length, and we have

$${\displaystyle {\text{length}}_{R}(M)={\text{length}}_{R}(L)+{\text{length}}_{R}(N)}$$

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

${\displaystyle 0=N_{0}\subsetneq N_{1}\subsetneq \cdots \subsetneq N_{n}=M}$

such that

${\displaystyle N_{i+1}/N_{i}{\text{ is simple for }}i=0,\dots ,n-1}$

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.

Finite dimensional vector spaces

Any finite dimensional vector space ${\displaystyle V}$ over a field ${\displaystyle k}$ has a finite length. Given a basis ${\displaystyle v_{1},\ldots ,v_{n}}$ there is the chain

$${\displaystyle 0\subset {\text{Span}}_{k}(v_{1})\subset {\text{Span}}_{k}(v_{1},v_{2})\subset \cdots \subset {\text{Span}}_{k}(v_{1},\ldots ,v_{n})=V}$$

which is of length ${\displaystyle n}$. It is maximal because given any chain,

$${\displaystyle V_{0}\subset \cdots \subset V_{m}}$$

the dimension of each inclusion will increase by at least ${\displaystyle 1}$. Therefore, its length and dimension coincide.

Artinian modules

Over a base ring ${\displaystyle R}$, 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]

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 ${\displaystyle f\in R(X)^{*}}$ on an algebraic variety. Given an algebraic variety ${\displaystyle X}$ and a subvariety ${\displaystyle V}$ of codimension 1^[3] the order of vanishing for a polynomial ${\displaystyle f\in R(X)}$ is defined as^[4]

$${\displaystyle \operatorname {ord} _{V}(f)={\text{length}}_{{\mathcal {O}}_{V,X}}\left({\frac {{\mathcal {O}}_{V,X}}{(f)}}\right)}$$

where ${\displaystyle {\mathcal {O}}_{V,X}}$ is the local ring defined by the stalk of ${\displaystyle {\mathcal {O}}_{X}}$ along the subvariety ${\displaystyle V}$^{[3] pages 426-227}, or, equivalently, the stalk of ${\displaystyle {\mathcal {O}}_{X}}$ at the generic point of ${\displaystyle V}$^{[5] page 22}. If ${\displaystyle X}$ is an affine variety, and ${\displaystyle V}$ is defined the by vanishing locus ${\displaystyle V(f)}$, then there is the isomorphism

$${\displaystyle {\mathcal {O}}_{V,X}\cong R(X)_{(f)}}$$

This idea can then be extended to rational functions ${\displaystyle F=f/g}$ on the variety ${\displaystyle X}$ where the order is defined as^[3]

$${\displaystyle \operatorname {ord} _{V}(F):=\operatorname {ord} _{V}(f)-\operatorname {ord} _{V}(g)}$$

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 ${\displaystyle Z(h)\subset \mathbb {P} ^{3}}$ defined by a polynomial ${\displaystyle h\in k[x_{0},x_{1},x_{2},x_{3}]}$, then the order of vanishing of a rational function

$${\displaystyle F={\frac {f}{g}}}$$

is given by

$${\displaystyle \operatorname {ord} _{Z(h)}(F)=\operatorname {ord} _{Z(h)}(f)-\operatorname {ord} _{Z(h)}(g)}$$

where

$${\displaystyle \operatorname {ord} _{Z(h)}(f)={\text{length}}_{{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}\left({\frac {{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}{(f)}}\right)}$$

For example, if ${\displaystyle h=x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{2}^{3}}$ and ${\displaystyle f=x^{2}+y^{2}}$ and ${\displaystyle g=h^{2}(x_{0}+x_{1}-x_{3})}$ then

$${\displaystyle \operatorname {ord} _{Z(h)}(f)={\text{length}}_{{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}\left({\frac {{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}{(x^{2}+y^{2})}}\right)=0}$$

since ${\displaystyle x^{2}+y^{2}}$ is a unit in the local ring ${\displaystyle {\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}$. In the other case, ${\displaystyle x_{0}+x_{1}-x_{3}}$ is a unit, so the quotient module is isomorphic to

$${\displaystyle {\frac {{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}{(h^{2})}}}$$

so it has length ${\displaystyle 2}$. This can be found using the maximal proper sequence

$${\displaystyle (0)\subset {\frac {{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}{(h)}}\subset {\frac {{\mathcal {O}}_{Z(h),\mathbb {P} ^{3}}}{(h^{2})}}}$$

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 function

$${\displaystyle {\frac {(z-1)^{3}(z-2)}{(z-1)(z-4i)}}}$$

has zeros of order 2 and 1 at ${\displaystyle 1,2\in \mathbb {C} }$ and a pole of order ${\displaystyle 1}$ at ${\displaystyle 4i\in \mathbb {C} }$. This kind of information can be encoded using the length of modules. For example, setting ${\displaystyle R(X)=\mathbb {C} [z]}$ and ${\displaystyle V=V(z-1)}$, there is the associated local ring ${\displaystyle {\mathcal {O}}_{V,X}}$ is ${\displaystyle \mathbb {C} [z]_{(z-1)}}$ and the quotient module

$${\displaystyle {\frac {\mathbb {C} [z]_{(z-1)}}{((z-4i)(z-1)^{2})}}}$$

Note that ${\displaystyle z-4i}$ is a unit, so this is isomorphic to the quotient module

$${\displaystyle {\frac {\mathbb {C} [z]_{(z-1)}}{((z-1)^{2})}}}$$

Its length is ${\displaystyle 2}$ since there is the maximal chain

$${\displaystyle (0)\subset {\frac {\mathbb {C} [z]_{(z-1)}}{((z-1))}}\subset {\displaystyle {\frac {\mathbb {C} [z]_{(z-1)}}{((z-1)^{2})}}}}$$

of submodules.^[6] More generally, using the Weierstrass factorization theorem a meromorphic function factors as

$${\displaystyle F={\frac {f}{g}}}$$

which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.