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( Link ). 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: Link 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.