Author: Olcay Coskun & Ergun Yalcin

Title: 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 $\varphi: D_{+} \to D^{+}$ 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 $$0 \to \widehat{\rm Ext}^{-1}_\gamma(\rho, D) \to D_{+} \maprt{\varphi} D^{+} \to \widehat{\rm Ext}^{0}_\gamma(\rho, D)\to 0$$ where $\rho$ denotes the restriction algebra and $\gamma$ 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.


Status: 

to appear J. Pure and Applied Algebra.