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

A Witt–burnside Ring Attached to a Pro-Dihedral Group

“…Work by the author as well as Y.T. Oh has begun to clarify the structure of these rings [Mil,Oh07,Oh09], but to date no detailed non-abelian examples appear in the literature. This article fills that gap.…”

“…Like the p-typical Witt vectors, for any field k, the Witt-Burnside ring W G (k) is always a local ring [Mil,Cor. 3.21], and when G is an infinite pro-p group W G (k) has characteristic 0 when k is a field of characteristic p [DS88, Cor 2, pg 115-116].…”

“…This comparison showed that W Z d p (k) for d > 1 lacks many of the nice properties enjoyed by the d = 1 case where one recovers the p-typical Witt vectors, i.e., a p-adically complete DVR. In particular, it is noted that unlike the p-typical Witt vectors, W Z d p (k) is not a noetherian ring and not even coherent 1 when d = 2 [Mil,Cor. 5.21].…”

“…5.21]. In fact, even the unique maximal ideal fails to be finitely generated, though in the d = 2 case, W Z 2 p (k) is reduced local ring of characteristic 0 [Mil,Thm. 6.12], an algebraic situation similar to rings of functions.…”

“…We also examine the nilpotent elements of W D 2 ∞ (k) where k has characteristic 2, and in contrast to W Z 2 p (k) for k of characteristic p, which is reduced [Mil,Theorem 6.12], there are nilpotent elements in W D 2 ∞ (k).…”