Gluing endo-permutation modules

Abstract

In this paper, I show that if p is an odd prime, and if P is a finite p-group, then there exists an exact sequence of abelian groups 0→ T (P )→ D(P )→ lim ←− 1<Q≤P D ( NP (Q)/Q )→ H1A≥2(P ),Z )(P ) , where D(P ) is the Dade group of P and T (P ) is the subgroup of endo-trivial modules. Here lim ←− 1<Q≤P D ( NP (Q)/Q ) is the group of sequences of compatible elements in the Dade groups D ( NP (Q)/Q ) for non trivial subgroups Q of P . The poset A≥2(P ) is the set of elementary abelian subgroups of rank at least 2 of P , ordered by inclusion. The group H A≥2(P ),Z )(P ) is the subgroup of H1A≥2(P ),Z ) consisting of classes of P -invariant 1-cocycles. A key result to prove that the above sequence is exact is a characterization of elements of 2D(P ) by sequences of integers, indexed by sections (T, S) of P such that T/S ∼= (Z/pZ), fulfilling certain conditions associated to subquotients of P which are either elementary abelian of rank 3, or extraspecial of order p and exponent p. AMS Subject classification : 20C20