Gluing Borel-Smith functions and the group of endo-trivial modules

Abstract

The aim of this paper is to describe the group of endo-trivial modules for a p-group P , in terms of the obstruction group for the gluing problem of Borel-Smith functions. Explicitly, we shall prove that there is a split exact sequenceof abelian groups where T (P ) is the endo-trivial group of P , and C b (P ) is the group of Borel-Smith functions on P . As a consequence, we obtain a set of generators of the group T (P ) that coincides with the relative syzygies found by Alperin. In order to prove the result, we solve gluing problems for the functor B * of super class functions, the functor R * Q of rational class functions and the functor C b of Borel-Smith functions. Contents

Obstructions for gluing biset functors

“…3 we have H −1 (D * G ; D Ω t ) = ker r G = 0, and Obs(D Ω t (G)) ∼ = S∈S Z/n S Zwhere n S is defined as above. By Proposition 7.2, we haveObs(R * Q (G)) ∼ = H 0 (A ≥2 (G)/G; Z) ∼ = S∈S Z.Hence we recover the obstruction group calculation Obs(C b (G)) = H 0 b (A ≥2 (G), pZ) G given in[8, Theorem 5.1].…”

“…For the torsion part of Dade group D t defined on p-groups with p odd, we recover the earlier obstruction group computations done by Bouc and Thévenaz [6]. We also recover the obstruction group calculations done by Coşkun [8] for the dual of the rational representation ring functor R * Q and for the Borel-Smith functor C b (Propositions 7.2 and 7.6). We also calculate the obstruction group Obs(D t (G)) for a 2-group G (see Proposition 7.3).…”

“…If F = B * is the biset functor of the dual of the Burnside ring, then by [8, Theorem 3.1] for any p-group G, we have ker r B * G = Hom ZD G (T, B * ) ∼ = Z, so ∂B * (G) is not zero. If F = R * Q is the biset functor of the dual of the rational representation ring, then Hom ZD G (T, R * Q ) = 0 if G is a p-group of rank at least 2, and it is equal to Z otherwise (see [8,Sec. 4…”

“…to give another proof for the rank calculation for the group of endo-trivial modules T (G) (see [8,Theorem 6.1]). Now let's assume that p is odd.…”

“…Hence as a direct consequence of Theorem 6.8, we obtain the following. Proposition 7.2 (Coşkun [8]). If G is p-group with rk(G) ≥ 2, then Obs(R * Q (G)) ∼ = H 0 (A ≥2 (G)/G; Z).…”