Endomorphism rings of finite length modules

Endomorphisms of finite length modules just move the factors around, right?

Fitting showed that the endomorphism ring of a finite length indecomposable module is very nice: it is a local ring with a nilpotent Jacobson radical. This is seen by looking at how it acts on the Loewy series of the module. Of course the ring itself can be only one-sided artinian, showing that the local ring need not be artinian. The paper MR784161PAMS 1985 shows that the other side of the endomorphism ring need not be artinian either.

This is somewhat nice to see because in my opinion the canonical decomposition of the endomorphism ring, and the most effective method of calculating the endomorphism ring concretely for finite dimensional algebras given by matrices, is to work down the Loewy series. This shows that the induced filtration on the endomorphism ring might be far from a composition series.

However, over finite dimensional algebras everything is just fine.

A family of cool examples are the group rings of a dihedral group of order 2⋅3n over a field of characteristic 3. It has two simple modules, and the corresponding PIMs have composition length 3n, with the composition factors alternating between the two simple modules. The endomorphism rings are k[x]/(x((3n−1)/2+1)). If p=3 is made more general, then we get the same result, with p−1=2.

Indeed, it is reasonably common that the endomorphism ring of a uniserial module is k[x]/(x^n), and one of the easiest examples to calculate is the endomorphism ring of the regular module over this ring.

The dihedral examples also show how the endomorphism ring can be blind to certain simple modules. The Loewy series is so regular, that endomorphisms cannot really distinguish between the −1s and the 1s in the composition series.

It also shows how the centralizer of the endomorphism ring can be smaller than the original ring itself: one loses a dimension, corresponding, I believe, to a missing −1 on the bottom. Of course the double centralizer property holds, it is just that the module has a non-trivial annihilator. Once both PIMs are included, one recovers the whole ring. I guess that is also cool, the second PIM (of dimension say 9) only contributes a single dimension to the ring itself. Very different from the semi-simple case.

The Brauer tree of the dihedral groups is very simple, just a •–• with one vertex exceptional of high degree.

It be nice to check out a larger (star-like) Brauer tree and see again explicitly how much simpler the endomorphism ring is than the original ring (it should again by k[x]/(x^n), with only one simple module, rather than one simple module for each branch of the star).