A universal group-theoretic characterisation of $p$-typical Witt vectors

TL;DR

This paper provides a universal group-theoretic characterization of p-typical Witt vectors that does not rely on ring structure, enabling broader application including non-commutative rings.

Contribution

It introduces a new group-theoretic characterization of Witt vectors that uniquely identifies them without using ring operations, suitable for non-commutative contexts.

Findings

Characterization holds for p ≠ 2

The property uniquely determines the Witt vector functor

Applicable to non-commutative rings

Abstract

For a prime $p$ and a commutative ring $R$ with unity, let $W(R)$ denote the group of $p$-typical Witt vectors. The group $W(R)$ is endowed with a Verschiebung operator $V: W(R)\to W(R)$ and a Teichm\"{u}ller map $\langle \ \rangle: R\rightarrow W(R)$. One of the properties satisfied by $V, \langle \ \rangle$ is that the map $R \to W(R)$ given by $x\mapsto V\langle x^p \rangle - p\langle x \rangle$ is an additive map. In this paper we show that for $p\neq 2$, this property essentially characterises the functor $W$. Unlike other characterisations, this is a group-theoretic characterisation, in the sense that it does not use the ring structure of $W(R)$. Most constructions of the group of $p$-typical Witt vectors of non-commutative rings do not have a ring structure, and hence the above characterisation is more suitable for generalisation to the non-commutative setup.