Fitting lemma
Encyclopedia
The Fitting lemma, named after the mathematician Hans Fitting
, is a basic statement in abstract algebra
. Suppose M is a module
over some ring
. If M is indecomposable
and has finite length
, then every endomorphism
of M is either bijective or nilpotent
.
As an immediate consequence, we see that the endomorphism ring
of every finite-length indecomposable module is local
.
A version of Fitting's lemma is often used in the representation theory of groups
. This is in fact a special case of the version above, since every K-linear representation of a group G can be viewed as a module over the group algebra
KG.
To prove Fitting's lemma, we take an endomorphism f of M and consider the following two sequences of submodules. The first sequence is the descending sequence im(f), im(f 2), im(f 3),..., the second sequence is the ascending sequence ker(f), ker(f 2), ker(f 3),.... Because M has finite length, the first sequence cannot be strictly decreasing forever, so there exists some n with im(f n) = im(f n+1). Likewise (as M has finite length) the second sequence cannot be strictly increasing forever, so there exists some m with ker(f m) = ker(f m+1). It is easily seen that im(f n) = im(f n+1) yields im(f n) = im(f n+1) = im(f n+2) = ..., and that ker(f m) = ker(f m+1) yields ker(f m) = ker(f m+1) = ker(f m+2) = ... . Putting k = max(m,n ), it now follows that im(f k) = im(f 2k) and ker(f k) = ker(f 2k). Hence,
(because every 
satisfies 
for some 
but also 
, so that 
, therefore 
and thus 
) and 
(since for every 
, there exists some 
such that 
(since 
), and thus 
, so that 
and thus 
). Consequently, M is the direct sum
of im(f k) and ker(f k). Because M is indecomposable, one of those two summands must be equal to M, and the other must be equal to {0}. Depending on which of the two summands is zero, we find that f is bijective or nilpotent.
Hans Fitting
Hans Fitting was a mathematician who worked in group theory...
, is a basic statement in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. Suppose M is a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over some ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
. If M is indecomposable
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...
and has finite length
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
, then every endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
of M is either bijective or nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
.
As an immediate consequence, we see that the endomorphism ring
Endomorphism ring
In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
of every finite-length indecomposable module is local
Local ring
In abstract algebra, more particularly 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 varieties or manifolds, or of algebraic number fields examined at a particular place, or...
.
A version of Fitting's lemma is often used in the representation theory of groups
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
. This is in fact a special case of the version above, since every K-linear representation of a group G can be viewed as a module over the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
KG.
To prove Fitting's lemma, we take an endomorphism f of M and consider the following two sequences of submodules. The first sequence is the descending sequence im(f), im(f 2), im(f 3),..., the second sequence is the ascending sequence ker(f), ker(f 2), ker(f 3),.... Because M has finite length, the first sequence cannot be strictly decreasing forever, so there exists some n with im(f n) = im(f n+1). Likewise (as M has finite length) the second sequence cannot be strictly increasing forever, so there exists some m with ker(f m) = ker(f m+1). It is easily seen that im(f n) = im(f n+1) yields im(f n) = im(f n+1) = im(f n+2) = ..., and that ker(f m) = ker(f m+1) yields ker(f m) = ker(f m+1) = ker(f m+2) = ... . Putting k = max(m,n ), it now follows that im(f k) = im(f 2k) and ker(f k) = ker(f 2k). Hence,
(because every satisfies for some but also , so that , therefore and thus ) and (since for every , there exists some such that (since ), and thus , so that and thus ). Consequently, M is the direct sumDirect sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
of im(f k) and ker(f k). Because M is indecomposable, one of those two summands must be equal to M, and the other must be equal to {0}. Depending on which of the two summands is zero, we find that f is bijective or nilpotent.