Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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.

share|improve this answer
    
thank you for your edit!But I think it has period 1 for p=2. –  trew Dec 17 '11 at 14:20
    
You're right - I should have said "period dividing 2" –  m_t Dec 17 '11 at 15:01
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.