A Tate cohomology sequence for generalized Burnside rings

We generalize the fundamental theorem for Burnside rings to the mark morphism of plus constructions defined by Boltje. The main observation is the following: If DD is a restriction functor for a finite group GG, then the mark morphism φ:D+→D+φ:D+→D+ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for GG) after composing with a suitable isomorphism of D+D+. As a consequence, we obtain an exact sequence of Mackey functors 0→Ext̂γ−1(ρ,D)→D+⟶φD+→Ext̂γ0(ρ,D)→0 where ρρ denotes the restriction algebra and γγ denotes the conjugation algebra for GG. Then, we show how one can calculate these Tate groups explicitly using group cohomology and give some applications to integrality conditions.