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algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, the length of a module over a ring R is a generalization of the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
which measures its size.Alt URL
/ref> 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. If R is an algebra over a field k, the length of a module is at most its dimension as a k-vector space. In
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
and
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a module over a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
commutative ring R can have finite length only when the module has
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
zero. Modules of finite length are finitely generated modules, but most finitely generated modules have infinite length. Modules of finite length are
Artinian module Artinian may refer to: Mathematics *Objects named for Austrian mathematician Emil Artin (1898–1962) **Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 **Artinian ring, a ring which satisfies ...
s and are fundamental to the theory of
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
s. 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 M be a (left or right) module over some ring R. Given a chain of submodules of M of the form :M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n, one says that n is the ''length'' of the chain. The ''length'' of M is the largest length of any of its chains. If no such largest length exists, we say that M has ''infinite length''. Clearly, if the length of a chain equals the length of the module, one has M_0=0 and M_n=M.


Length of a ring

The length of a ring R is the length of the longest chain of ideals; that is, the length of R considered as a module over itself by left multiplication. By contrast, the
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally ...
of R is the length of the longest chain of ''prime'' ideals.


Properties


Finite length and finite modules

If an R-module M 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 R-module M has finite length if and only if it is both a
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the pr ...
and an
Artinian module Artinian may refer to: Mathematics *Objects named for Austrian mathematician Emil Artin (1898–1962) **Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 **Artinian ring, a ring which satisfies ...
(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

Suppose0\rarr L \rarr M \rarr N \rarr 0is a
short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
of R-modules. Then M has finite length if and only if ''L'' and ''N'' have finite length, and we have \text_R(M) = \text_R(L) + \text_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 :0=N_0\subsetneq N_1 \subsetneq \cdots \subsetneq N_n=M such that :N_/N_i \texti=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''.


Examples


Finite dimensional vector spaces

Any finite dimensional vector space V over a field k has a finite length. Given a basis v_1,\ldots,v_n there is the chain0 \subset \text_k(v_1) \subset \text_k(v_1,v_2) \subset \cdots \subset \text_k(v_1,\ldots, v_n) = Vwhich is of length n. It is maximal because given any chain,V_0 \subset \cdots \subset V_mthe dimension of each inclusion will increase by at least 1. Therefore, its length and dimension coincide.


Artinian modules

Over a base ring R,
Artinian module Artinian may refer to: Mathematics *Objects named for Austrian mathematician Emil Artin (1898–1962) **Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 **Artinian ring, a ring which satisfies ...
s 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 \mathbb/n\mathbb (viewed as a module over the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s Z) is equal to the number of prime factors of n, with multiple prime factors counted multiple times. This follows from the fact that the submodules of \mathbb/n\mathbb are in one to one correspondence with the positive divisors of n, this correspondence resulting itself from the fact that \Z 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 f \in R(X)^* on an algebraic variety. Given an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
X and a subvariety V of
codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals ...
1 the order of vanishing for a polynomial f \in R(X) is defined as\operatorname_V(f) = \text_\left( \frac \right)where \mathcal_ is the local ring defined by the stalk of \mathcal_X along the subvariety V pages 426-227, or, equivalently, the stalk of \mathcal_X at the generic point of V page 22. If X is an affine variety, and V is defined the by vanishing locus V(f), then there is the isomorphism\mathcal_ \cong R(X)_This idea can then be extended to
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s F = f/g on the variety X where the order is defined as\operatorname_V(F) := \operatorname_V(f) - \operatorname_V(g) which is similar to defining the order of zeros and poles in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
.


Example on a projective variety

For example, consider a projective surface Z(h) \subset \mathbb^3 defined by a polynomial h \in k _0,x_1,x_2,x_3/math>, then the order of vanishing of a rational functionF = \fracis given by\operatorname_(F) = \operatorname_(f) - \operatorname_(g) where\operatorname_(f) = \text_\left( \frac \right)For example, if h = x_0^3 + x_1^3 + x_2^3 + x_2^3 and f = x^2 + y^2 and g = h^2(x_0 + x_1 - x_3) then\operatorname_(f) = \text_\left( \frac \right) = 0since x^2 + y^2 is a unit in the
local ring In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
\mathcal_. In the other case, x_0 + x_1 - x_3 is a unit, so the quotient module is isomorphic to\fracso it has length 2. This can be found using the maximal proper sequence(0) \subset \frac \subset \frac


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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. For example, the function\frachas zeros of order 2 and 1 at 1, 2 \in \mathbb and a pole of order 1 at 4i \in \mathbb. This kind of information can be encoded using the length of modules. For example, setting R(X) = \mathbb /math> and V = V(z-1), there is the associated local ring \mathcal_ is \mathbb and the quotient module \fracNote that z-4i is a unit, so this is isomorphic to the quotient module\fracIts length is 2 since there is the maximal chain(0) \subset \frac \subset of submodules. More generally, using the
Weierstrass factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its Zero of a function, zeroes. The theorem m ...
a meromorphic function factors asF = \fracwhich 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 In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its Zero of a function, zeroes. The theorem m ...
* Serre's multiplicity conjectures * Hilbert scheme - can be used to study modules on a scheme with a fixed length * Krull–Schmidt theorem


References


External links

*Steven H. Weintraub, ''Representation Theory of Finite Groups'' AMS (2003) , {{isbn, 978-0-8218-3222-6 *Allen Altman, Steven Kleiman,
A term of commutative algebra
'. *The Stacks project
''Length''
Module theory