What do we know about periodic modules in p-groups? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T21:56:28Z http://mathoverflow.net/feeds/question/83520 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83520/what-do-we-know-about-periodic-modules-in-p-groups What do we know about periodic modules in p-groups? trew 2011-12-15T13:40:07Z 2011-12-28T15:46:12Z <p>Hi,</p> <p>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( <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.ijm/1255989005" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.ijm/1255989005</a> ). 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:</p> <p>In which dimensions can a module of period n occur?(results like in this paper: <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.ijm/1256048241" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.ijm/1256048241</a> where it is proven that a power of p divides the dimension)</p> <p>Which periods can occur in a given group?</p> <p>Is there any interesting relation of the subcategory of periodic modules and the pure group structure?</p> <p>Thank you</p> <p>edit: Another question: Can we give an example of a periodic module in an arbitrary KG?Maybe there is a canonical construction.</p> <p>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</p> <p>$M_E$ has maximal complextity 1 for an elementar abelian subgroup E of G iff</p> <p>M is a direct sum of indecomposable periodcis and projectives iff</p> <p>in the minimal projective resolution the terms have bounded dimension.</p> http://mathoverflow.net/questions/83520/what-do-we-know-about-periodic-modules-in-p-groups/83525#83525 Answer by mt for What do we know about periodic modules in p-groups? mt 2011-12-15T14:22:44Z 2011-12-17T13:27:25Z <p>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.</p> <p>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. </p> <p>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.</p> <p>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.</p>