Editing Length of a module

{{Short description|In algebra, integer associated to a module}}
In [[algebra]], the '''length''' of a [[module (mathematics)|module]] over a ring <math>R</math> is a generalization of the [[dimension (vector space)|dimension]] of a [[vector space]] which measures its size.<ref name=":0">{{Cite web|title=A Term of Commutative Algebra|url=http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html|website=www.centerofmathematics.com|pages=153–158|url-status=live|archive-url=https://web.archive.org/web/20130302125321/http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html|archive-date=2013-03-02|access-date=2020-05-22}} [https://www.mi.fu-berlin.de/en/math/groups/arithmetic_geometry/teaching/exercises/Altman_-Kleiman---A-term-of-commutative-algebra-_2017_.pdf Alt URL]</ref> <sup>page 153</sup> It is defined to be the length of the longest chain of [[submodule]]s. 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 ring|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 module]]s, but most finitely generated modules have infinite length. Modules of finite length are [[Artinian module]]s and are fundamental to the theory of [[Artinian ring]]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 [[General position|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 (mathematics)|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.<ref name=":0" /> The ''length'' of <math>M</math> is the largest length of any of its chains. If no such largest length exists, we say that <math>M</math> has ''infinite length''. Clearly, if the length of a chain equals the length of the module, one has <math>M_0=0</math> and <math>M_n=M.</math>

=== Length of a ring ===
The length of a ring <math>R</math> is the length of the longest chain of [[Ideal (ring theory)|ideals]]; that is, the length of <math>R</math> considered as a module over itself by left multiplication. By contrast, the [[Krull dimension]] of <math>R</math> is the length of the longest chain of [[Prime ideal|''prime'' ideals]].

== Properties ==

=== Finite length and finite modules ===
If an <math>R</math>-module <math>M</math> has finite length, then it is [[finitely generated module|finitely generated]].<ref>{{Cite web|title=Lemma 10.51.2 (02LZ)—The Stacks project|url=https://stacks.math.columbia.edu/tag/02LZ|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref> 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]]<ref name=":0" /> (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<math display=block>0\rarr L \rarr M \rarr N \rarr 0</math>is a [[short exact sequence]] of <math>R</math>-modules. Then M has finite length if and only if ''L'' and ''N'' have finite length, and we have <math display=block>\text{length}_R(M) = \text{length}_R(L) + \text{length}_R(N)</math> 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 ===
{{main|Jordan–Hölder theorem}}
A [[composition series]] of the module ''M'' is a chain of the form

:<math>0=N_0\subsetneq N_1 \subsetneq \cdots \subsetneq N_n=M</math>

such that

:<math>N_{i+1}/N_i \text{ is simple for }i=0,\dots,n-1</math>

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 <math>V</math> over a field <math>k</math> has a finite length. Given a basis <math>v_1,\ldots,v_n</math> there is the chain<math display=block>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</math>which is of length <math>n</math>. It is maximal because given any chain,<math display=block>V_0 \subset \cdots \subset V_m</math>the dimension of each inclusion will increase by at least <math>1</math>. Therefore, its length and dimension coincide.

=== Artinian modules ===
Over a base ring <math>R</math>, [[Artinian module]]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]].<ref name=":1">{{Cite book|last=Fulton, William, 1939-|url=https://www.worldcat.org/oclc/38048404|title=Intersection theory|date=1998|publisher=Springer|isbn=3-540-62046-X|edition= 2nd|location=Berlin|pages=8–10|oclc=38048404}}</ref>

==== Zero module ====
The zero module is the only one with length 0.

==== Simple modules ====
Modules with length 1 are precisely the [[simple module]]s.

==== Artinian modules over Z ====
The length of the [[cyclic group]] <math>\mathbb{Z}/n\mathbb{Z}</math> (viewed as a module over the [[integer]]s '''Z''') is equal to the number of [[prime number|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==
{{Main|Intersection multiplicity}}
For the needs of [[intersection theory]], [[Jean-Pierre Serre]] introduced a general notion of the [[multiplicity (mathematics)|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 {{mvar|n}} [[hypersurface|algebraic hypersurface]]s in a {{mvar|n}}-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 ===
{{technical|section and subsections|date=May 2020}}
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 (mathematics)|subvariety]] <math>V</math> of [[codimension]] 1<ref name=":1" /> the order of vanishing for a polynomial <math>f \in R(X)</math> is defined as<ref>{{Cite web|title=Section 31.26 (0BE0): Weil divisors—The Stacks project|url=https://stacks.math.columbia.edu/tag/0BE0|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref><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><ref name=":1" /> <sup>pages 426-227</sup>, or, equivalently, the [[Stalk of a sheaf|stalk]] of <math>\mathcal{O}_X</math> at the generic point of <math>V</math><ref>{{Cite book|last=Hartshorne|first=Robin|url=http://link.springer.com/10.1007/978-1-4757-3849-0|title=Algebraic Geometry|date=1977|publisher=Springer New York|isbn=978-1-4419-2807-8|series=Graduate Texts in Mathematics|volume=52|location=New York, NY|doi=10.1007/978-1-4757-3849-0|s2cid=197660097 }}</ref> <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 function]]s <math>F = f/g</math> on the variety <math>X</math> where the order is defined as<ref name=":1" /><math display=block>\operatorname{ord}_V(F) := \operatorname{ord}_V(f) - \operatorname{ord}_V(g) </math> 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]] <math>Z(h) \subset \mathbb{P}^3</math> defined by a polynomial <math>h \in k[x_0,x_1,x_2,x_3]</math>, then the order of vanishing of a rational function<math display=block>F = \frac{f}{g}</math>is given by<math display=block>\operatorname{ord}_{Z(h)}(F) = \operatorname{ord}_{Z(h)}(f) - \operatorname{ord}_{Z(h)}(g) </math>where<math display=block>\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)</math>For example, if <math>h = x_0^3 + x_1^3 + x_2^3 + x_2^3</math> and <math>f = x^2 + y^2</math> and <math>g = h^2(x_0 + x_1 - x_3)</math> then<math display=block>\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</math>since <math>x^2 + y^2</math> is a [[Unit (ring theory)|unit]] in the [[local ring]] <math>\mathcal{O}_{Z(h),\mathbb{P}^3}</math>. In the other case, <math>x_0 + x_1 - x_3</math> is a unit, so the quotient module is isomorphic to<math display=block>\frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h^2)}</math>so it has length <math>2</math>. This can be found using the maximal proper sequence<math display=block>(0) \subset \frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h)} \subset \frac{\mathcal{O}_{Z(h), \mathbb{P}^3}}{(h^2)}</math>

==== Zero and poles of an analytic function ====
The order of vanishing is a generalization of the order of zeros and poles for [[meromorphic function]]s in [[complex analysis]]. For example, the function<math display=block>\frac{(z-1)^3(z-2)}{(z-1)(z-4i)}</math>has zeros of order 2 and 1 at <math>1, 2 \in \mathbb{C}</math> and a pole of order <math>1</math> at <math>4i \in \mathbb{C}</math>. This kind of information can be encoded using the length of modules. For example, setting <math>R(X) = \mathbb{C}[z]</math> and <math>V = V(z-1)</math>, there is the associated local ring <math>\mathcal{O}_{V,X}</math> is <math>\mathbb{C}[z]_{(z-1)}</math> and the quotient module <math display=block>\frac{\mathbb{C}[z]_{(z-1)}}{((z-4i)(z-1)^2)}</math>Note that <math>z-4i</math> is a unit, so this is isomorphic to the quotient module<math display=block>\frac{\mathbb{C}[z]_{(z-1)}}{((z-1)^2)}</math>Its length is <math>2</math> since there is the maximal chain<math display=block>(0) \subset \frac{\mathbb{C}[z]_{(z-1)}}{((z-1))} \subset {\displaystyle {\frac {\mathbb {C} [z]_{(z-1)}}{((z-1)^{2})}}}</math>of submodules.<ref>{{Cite web|title=Section 10.120 (02MB): Orders of vanishing—The Stacks project|url=https://stacks.math.columbia.edu/tag/02MB|website=stacks.math.columbia.edu|access-date=2020-05-22}}</ref> 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 (mathematics)|scheme]] with a fixed length
*[[Krull–Schmidt theorem]]

== References ==
{{Reflist}}

== External links ==

*Steven H. Weintraub, ''Representation Theory of Finite Groups'' AMS (2003) {{isbn|0-8218-3222-0}}, {{isbn|978-0-8218-3222-6}}
*Allen Altman, Steven Kleiman, ''[http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html A term of commutative algebra]''. 
*The Stacks project. [https://stacks.math.columbia.edu/tag/00IU ''Length'']

[[Category:Module theory]]