Length of a module explained
In algebra, the length of a module over a ring
is a generalization of the
dimension of a
vector space which measures its size.
[1] 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
is an algebra over a field
, the length of a module is at most its dimension as a
-vector space.
In commutative algebra and algebraic geometry, a module over a Noetherian commutative ring
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 called
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
be a (left or right) module over some
ring
. Given a chain of submodules of
of the form
M0\subsetneqM1\subsetneq … \subsetneqMn,
one says that
is the
length of the chain. 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
Noetherian module 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 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
See main article: Jordan–Hölder theorem. A composition series of the module M is a chain of the form
0=N0\subsetneqN1\subsetneq … \subsetneqNn=M
such that
Ni+1/Niissimplefori=0,...,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
over a field
has a finite length. Given a basis
there is the chain
which 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] 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
(viewed as a module over the
integers
Z) 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
See main article: Intersection multiplicity. 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
on an algebraic variety. Given an
algebraic variety
and a subvariety
of
codimension 1 the order of vanishing for a polynomial
is defined as
[4] where
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
[5] page 22. If
is an
affine variety, and
is defined the by vanishing locus
, then there is the isomorphism
This idea can then be extended to
rational functions
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
defined by a polynomial
, then the order of vanishing of a rational function
is given by
where
For example, if
and
and
then
since
is a
unit in the
local ring
. In the other case,
is a unit, so the quotient module is isomorphic to
so 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 module
Its length is
since there is the maximal chain
of submodules.
[6] More generally, using the
Weierstrass factorization theorem a meromorphic function factors as
which is a (possibly infinite) product of linear polynomials in both the numerator and denominator.
See also
External links
Notes and References
- Web site: A Term of Commutative Algebra. www.centerofmathematics.com. 153–158. live. https://web.archive.org/web/20130302125321/http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html. 2013-03-02. 2020-05-22. Alt URL
- Web site: Lemma 10.51.2 (02LZ)—The Stacks project. stacks.math.columbia.edu. 2020-05-22.
- Book: Fulton, William, 1939-. Intersection theory. 1998. Springer. 3-540-62046-X. 2nd. Berlin. 8–10. 38048404.
- Web site: Section 31.26 (0BE0): Weil divisors—The Stacks project. stacks.math.columbia.edu. 2020-05-22.
- Book: Hartshorne, Robin. Algebraic Geometry. 1977. Springer New York. 978-1-4419-2807-8. Graduate Texts in Mathematics. 52. New York, NY. 10.1007/978-1-4757-3849-0. 197660097 .
- Web site: Section 10.120 (02MB): Orders of vanishing—The Stacks project. stacks.math.columbia.edu. 2020-05-22.