# The Roquette category of finite $p$-groups

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 11, page 2843-2886
- ISSN: 1435-9855

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topBouc, 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 -

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