Borel–Smith functions and the Dade group

Abstract

We show that there is an exact sequence of biset functors over p-groupswhere C b is the biset functor for the group of Borel-Smith functions, B * is the dual of the Burnside ring functor, D Ω 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 × of units of the Burnside ring and obtain exact sequences involving the torsion part of D Ω , mod 2 reduction of C b , and B × .

Tornehave morphisms, II: The lifted Tornehave morphism and the dual of the Burnside functor

“…We write mod to denote the morphism of biset functors whose coordinate map mod G : B * (G) → B × (G) is given by reduction from coefficients in Z to coefficients in F 2 . An explicit treatment of the lifted tom Dieck morphism die(), as a morphism of biset functors, appears in Bouc-Yalçın [11,Section 3]. Its coordinate map die G : A R (G) → B * (G) goes back to tom Dieck [12,Section III.5].…”

“…Later in this section, we shall define the lifted exponential morphism exp() by means of a formula but, in Section 4, we shall find that exp = die • lin. Thus, everything in the left-hand diagram above is already implicit in [11].…”

Tornehave morphisms III: The reduced Tornehave morphism and the Burnside unit functor

“…The linearization morphism lin : B → A R , the reduced tom Dieck morphism die : A R → B × and the reduced exponential morphism exp = die • lin are reviewed in [3] and [4]. These three morphisms of biset functors are also discussed in Bouc and Yalçın [11] and other papers cited therein. The reduced Tornehave morphism is an inflaky morphism torn π : K → B × , where K = Ker(lin).…”

Alcahestic subalgebras of the alchemic algebra and a correspondence of simple modules

“…For an improvement of this result to Z, see [7] and for more exact sequences relating these functors, see [11]. Some other well-known examples of biset functors are the functor of units of the Burnside ring [5] and the functor of the group of relative syzygies [6].…”