The Roquette category of finite $p$-groups

Abstract

Let $p$ be a prime number. This paper introduces the Roquette category ${\mathcal{R}}_p$ of finite $p$-groups, which is an additive tensor category containing all finite $p$-groups among its objects. In ${\mathcal{R}}_p$, every finite $p$-group $P$ admits a canonical direct summand $\partial P$, called the edge of $P$. Moreover $P$ splits uniquely as a direct sum of edges of Roquette $p$-groups, and the tensor structure of ${\mathcal{R}}_p$ can be described in terms of such edges.

The main motivation for considering this category is that the additive functors from ${\mathcal{R}}_p$ to abelian groups are exactly the rational $p$-biset functors. This yields in particular very efficient ways of computing such functors on arbitrary $p$-groups: this applies to the representation functors $R_K$, where $K$ is any field of characteristic 0, but also to the functor of units of Burnside rings, or to the torsion part of the Dade group.