In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, the length of a
module 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 coordin ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
which measures its size.
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/ref> page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type.
Related concepts in ...
s. It is defined to be the length of the longest chain of submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the m ...
s. Modules with ''finite'' length share many important properties with finite-dimensional vector spaces.
Other concepts used to 'count' in ring and module theory
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
are depth and height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory
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 coordin ...
whereas length is used to analyze finite modules. There are also various ideas of ''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 coordin ...
'' that are useful. Finite length commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s play an essential role in functorial treatments of formal algebraic geometry and deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesima ...
where Artin rings are used extensively.
Definition
Length of a module
Let be a (left or right) module over some ring . Given a chain of submodules of of the form
:
we say that is the length of the chain. The length of is defined to be the largest length of any of its chains. If no such largest length exists, we say that has infinite length.
Length of a ring
A ring is said to have finite length as a ring if it has finite length as a left -module.
Properties
Finite length and finite modules
If an -module 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 -module 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 prop ...
and an Artinian module (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
Supposeis a short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the contex ...
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.
Jordan–Hölder theorem
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''.
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
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem ...
.
Zero module
The zero module is the only one with length 0.
Simple modules
Modules with length 1 are precisely the simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
s.
Artinian modules over Z
The length of the cyclic group (viewed as a module over the integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s Z) is equal to the number of prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
factors of , with multiple prime factors counted multiple times. This can be found by using the Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...
.
Use in multiplicity theory
For the need of Intersection theory
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem ...
, Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the i ...
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
Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
that asserts that the sum of the multiplicities of the intersection points of algebraic hypersurfaces in a -dimensional projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
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 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 set of solutions of a system of polynomial equations over the real or complex numbers. ...
and a subvariety
A subvariety (Latin: ''subvarietas'') in botanical nomenclature is a taxonomic rank. They are rarely used to classify organisms.
Plant taxonomy
Subvariety is ranked:
*below that of variety (''varietas'')
*above that of form (''forma'').
Subvari ...
of codimension 1 the order of vanishing for a polynomial is defined aswhere is the local ring defined by the stalk of along the subvariety pages 426-227, or, equivalently, the stalk of at the generic point of 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 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 on the variety where the order is defined as 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