A universal group-theoretic characterisation of $p$-typical Witt vectors
Supriya Pisolkar, Biswanath Samanta
TL;DR
This work provides a universal, group-theoretic characterisation of the functor of $p$-typical Witt vectors $W$ for $p\neq 2$, avoiding reliance on the ring structure and aiming at noncommutative generalisation. It builds and contrasts two parallel constructions: the established Cuntz–Deninger functor $E$ and a new universal weak pre-Witt functor $C$, proving $C\cong E$ and that $W$ is the universal pre-Witt functor satisfying a fixed set of natural properties. A key technical achievement is showing that for polynomial rings, $C(A)\cong E(A)\cong X(A)$ and that the induced maps become isomorphisms, enabling a passage from polynomial algebras to general commutative rings via presentations. The results collectively provide a robust, ring-structure-free characterisation of $W$ that extends naturally toward noncommutative contexts, with potential for broader generalisation of Witt-vector-type functors.
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 Teichmüller 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.