Fitting lemma

Summary

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 an automorphism or nilpotent.[1]

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.

ProofEdit

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  ,
  • the second sequence is the ascending sequence  

Because   has finite length, both of these sequences must eventually stabilize, so there is some   with   for all  , and some   with   for all  .

Let now  , and note that by construction   and  .

We claim that  . Indeed, every   satisfies   for some   but also  , so that  , therefore   and thus  .

Moreover,  : for every  , there exists some   such that   (since  ), and thus  , so that   and thus  .

Consequently,   is the direct sum of   and  . (This statement is also known as the Fitting decomposition theorem.[2]) Because   is indecomposable, one of those two summands must be equal to  , and the other must be the trivial submodule. Depending on which of the two summands is zero, we find that   is either bijective or nilpotent.[3]

NotesEdit

  1. ^ Jacobson, A lemma before Theorem 3.7.
  2. ^ "Fitting's lemma". PlanetMath. Retrieved 22 July 2022.
  3. ^ Jacobson (2009), p. 113–114.

ReferencesEdit