Borel–Smith functions and the Dade group

We show that there is an exact sequence of biset functors over p-groups0→Cb→jB∗→ΨDΩ→0 where CbCb is the biset functor for the group of Borel–Smith functions, B∗B∗ is the dual of the Burnside ring functor, DΩDΩ 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 [S. Bouc, A remark on the Dade group and the Burnside group, J. Algebra 279 (2004) 180–190]. We also show that the kernel of mod 2 reduction of Ψ   is naturally equivalent to the functor B×B× of units of the Burnside ring and obtain exact sequences involving the torsion part of DΩDΩ, mod 2 reduction of CbCb, and B×B×.