A Tate cohomology sequence for generalized Burnside rings

Abstract

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 D is a restriction functor for a finite group G, then the mark morphism φ : D+ → D+ is the same as the norm map of the Tate cohomology sequence (over conjugation algebra for G) after composing with a suitable isomorphism of D+. As a consequence, we obtain an exact sequence of Mackey functors 0 → over(Ext, ̂)γ - 1 (ρ, D) → D+ over({long rightwards arrow}, φ) D+ → over(Ext, ̂)γ 0 (ρ, D) → 0 where ρ denotes the restriction algebra and γ denotes the conjugation algebra for G. Then, we show how one can calculate these Tate groups explicitly using group cohomology and give some applications to integrality conditions. © 2008 Elsevier B.V. All rights reserved.