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

TL;DR

This paper investigates the ring structure of Witt-Burnside rings W_G(k) for pro-p groups G and fields k of characteristic p, revealing complexities beyond the classical Witt vector case.

Contribution

It provides a detailed analysis of the structure of Witt-Burnside rings for various pro-p groups, highlighting their increased complexity compared to classical Witt vectors.

Findings

Witt-Burnside rings exhibit more complex structures than classical Witt vectors.

The structure varies significantly depending on the pro-p group G.

Classical Witt vectors are a special case within this broader framework.

Abstract

The classical 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 W_G attached to any profinite group G. The classical Witt vectors are those attached to the p-adic integers. Here we examine the ring structure of 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 classical case.