A functorial presentation of units of Burnside rings

TL;DR

This paper characterizes the units of Burnside rings as a functor, providing a projective resolution, describing their structure via limits over specific sections of groups, and revealing their non-finite generation and complexity.

Contribution

It offers a functorial presentation of units of Burnside rings, explicit generators for related kernels, and insights into their algebraic and categorical properties.

Findings

Provides a projective resolution of the dual functor ^ imes.

Describes B^ imes(G) as a limit over specific group sections.

Shows B^ imes is not finitely generated, but its dual is finitely generated.

Abstract

Let $B^\times$ be the biset functor over $\mathbb{F}_2$ sending a finite group~$G$ to the group $B^\times(G)$ of units of its Burnside ring $B(G)$, and let $\widehat{B^\times}$ be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from $B^\times$ in the dual Burnside functor $\widehat{\mathbb{F}_2B}$, or equivalently, an explicit set of generators $\mathcal{G}_S$ of the kernel $L$ of the natural surjection $\mathbb{F}_2B\to \widehat{B^\times}$. This yields a two terms projective resolution of $\widehat{B^\times}$, leading to some information on the extension functors $\mathrm{Ext}^1(-,B^\times)$. For a finite group $G$, this also allows for a description of $B^\times(G)$ as a limit of groups $B^\times(T/S)$ over sections $(T,S)$ of $G$ such that $T/S$ is cyclic of odd prime order, Klein four, dihedral of order 8, or a Roquette 2-group. Another consequence is that the biset functor $B^\times$ is not finitely generated, and that its dual $\widehat{B^\times}$ is finitely generated, but not finitely presented. The last result of the paper shows in addition that $\mathcal{G}_S$ is a minimal set of generators of $L$, and it follows that the lattice of subfunctors of $L$ is uncountable.