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Journal of the London Mathematical Society Advance Access published online on July 8, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn039
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© 2008 London Mathematical Society

Gluing torsion endo-permutation modules

Serge Bouc

CNRS-LAMFA
Université de Picardie – Jules Verne
33, rue St Leu
F-80039 Amiens cedex 1
France
serge.bouc@u-picardie.fr

Jacques Thévenaz

Institut de Géométrie Algèbre et Topologie
EPFL
Bâtiment BCH
CH-1015 Lausanne
Switzerland

Let k be a field of characteristic p, and let P be a finite p-group, where p is an odd prime. In this paper, we consider the problem of gluing compatible families of endo-permutation modules: being given a torsion element MQ in the Dade group D(NP(Q)/Q), for each non-trivial subgroup Q of P, subject to obvious compatibility conditions, we show that it is always possible to find an element M in the Dade group of P such that Formula for all Q, but that M need not be a torsion element of D(P). The obstruction to this is controlled by an element in the zeroth cohomology group over F2 of the poset of elementary abelian subgroups of P of rank at least 2. We also give an example of a similar situation, when MQ is only given for centric subgroups Q of P. Moreover, general results about biset functors and the Dade functor are given in two appendices.

2000 Mathematics Subject Classification 20C20.

Received June 9, 2006; revised March 11, 2008;
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