In mathematics, 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.
To prove Fitting's lemma, we take an endomorphism f of M and consider the following two chains of submodules:
- The first is the descending chain
,
- the second is the ascending chain
![{\displaystyle \mathrm {ker} (f)\subseteq \mathrm {ker} (f^{2})\subseteq \mathrm {ker} (f^{3})\subseteq \ldots }](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8ad25462b2b607488441804eb7df3275819297b7)
Because
has finite length, both of these chains 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.) Because
is indecomposable, one of those two summands must be equal to
and the other must be the zero submodule. Depending on which of the two summands is zero, we find that
is either bijective or nilpotent.[2]
- ^ Jacobson 2009, A lemma before Theorem 3.7.
- ^ Jacobson (2009), p. 113–114.