Borel-smith Functions and the Dade Group

We show that there is an exact sequence of biset functors over p-groups 0 → Cb j −→B∗ Ψ −→D → 0 where Cb is the biset functor for the group of Borel-Smith functions, B ∗ is the dual of the Burnside ring functor, D is the functor for the subgroup of the Dade group generated by relative syzygies, and the natural transformation Ψ is the transformation recently introduced by the first author in [5]. We also show that the kernel of mod 2 reduction of Ψ is naturally equivalent to the functor B× of units of the Burnside ring and obtain exact sequences involving the torsion part of D, mod 2 reduction of Cb, and B ×.