The Roquette category of finite $p$-groups

Bouc, Serge. "The Roquette category of finite $p$-groups." Journal of the European Mathematical Society 017.11 (2015): 2843-2886. <http://eudml.org/doc/277769>.

@article{Bouc2015,
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.},
author = {Bouc, Serge},
journal = {Journal of the European Mathematical Society},
keywords = {$p$-group; Roquette; rational; biset; genetic subgroups; -group; Roquette; rational biset functor; biset; genetic subgroups},
language = {eng},
number = {11},
pages = {2843-2886},
publisher = {European Mathematical Society Publishing House},
title = {The Roquette category of finite $p$-groups},
url = {http://eudml.org/doc/277769},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Bouc, Serge
TI - The Roquette category of finite $p$-groups
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 11
SP - 2843
EP - 2886
AB - 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.
LA - eng
KW - $p$-group; Roquette; rational; biset; genetic subgroups; -group; Roquette; rational biset functor; biset; genetic subgroups
UR - http://eudml.org/doc/277769
ER -