Question

Hi,

a module in KG,where G is a p-group and K a field of characteristic p, is called periodic if $ \Omega^{n} M = M $, for a natural n. In general the full subcategory of periodic modules seems to have also wild representation type( http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1255989005 ). I wonder if there are still some interesting results about periodic modules. So I search for a kind of up-to-date survey paper listing such results. some questions are:

In which dimensions can a module of period n occur?(results like in this paper: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ijm/1256048241 where it is proven that a power of p divides the dimension)

Which periods can occur in a given group?

Is there any interesting relation of the subcategory of periodic modules and the pure group structure?

Thank you

edit: Another question: Can we give an example of a periodic module in an arbitrary KG?Maybe there is a canonical construction.

edit2: after reading parts of benson im a bit confused.for example in the introduction he says compelextity 1 is equivalent being periodic.But he says something else in a later theorem. Is the following correct?: M has complextity 1 iff

$M_E $ has maximal complextity 1 for an elementar abelian subgroup E of G iff

M is a direct sum of indecomposable periodcis and projectives iff

in the minimal projective resolution the terms have bounded dimension.

Answer 1

One interesting result on which periods can occur is that if $\operatorname{Ext}^*_{kG}(k,k)$ is finitely generated over a subring generated by elements of degree at most $m$, then any periodic $kG$-module has period at most $m$. You can find this result (and many other relevant ones) in Benson's book Representations and Cohomology vol 2: that and the references given there would be a good place to start.

Okuyama and Sasaki's "Periodic modules of large periods for metacyclic p-groups" (J.Algebra 144) might interest you. I wrote a paper about which $p$-groups can have periodic modules of dimension $p$ (the smallest possible dimension) and what the periods are, called `Periodic modules of dimension p' in Quarterly Journal of Mathematics 61 no. 3.

So far as I know, the answers to questions like "which periods can occur in a given group?", and "in which dimensions can a module of period n occur?" are not known except for in specific cases.

In response to your edit: if $G$ is a $p$-group, choose $H \leq G$ of order $p$. There's an exact sequence $0\to k \to kH \to kH \to k \to 0$, induce it to $G$. The middle terms are projective, so $k\uparrow^G$ is periodic of period two.