Fitting lemma

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 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.[2]

NotesEdit

  1. ^ Jacobson, A lemma before Theorem 3.7.
  2. ^ Jacobson (2009), p. 113–114.

ReferencesEdit

  • Jacobson, Nathan (2009), Basic algebra, 2 (2nd ed.), Dover, ISBN 978-0-486-47187-7