The Roquette category of finite p-groups

Let p be a prime number. This paper introduces the Roquette category Rp of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In Rp, every finite p-group P admits a canonical direct summand ∂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 Rp can be described in terms of such edges. The main motivation for considering this category is that the additive functors fromRp 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 RK , 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. AMS Subject classification : 18B99, 19A22, 20C99, 20J15.