On the structure of Witt–Burnside rings attached to pro-$p$ groups

Abstract

The $p$-typical Witt vectors are a ubiquitous object in algebra and number theory. They arise as a functorial construction that takes perfect fields $k$ of prime characteristic $p>0$ to $p$-adically complete discrete valuation rings of characteristic 0 with residue field $k$ and are universal in that sense. A. Dress and C. Siebeneicher generalized this construction by producing a functor ${\mathbf{W}}_G$ attached to any profinite group $G$. The $p$-typical Witt vectors arise as those attached to the $p$-adic integers. Here we examine the ring structure of ${\mathbf{W}}_G(k)$ for several examples of pro-$p$ groups $G$ and fields $k$ of characteristic $p$. We will show that the structure is surprisingly more complicated than the $p$-typical case.