A universal construction of $p$-typical Witt vectors of associative rings

TL;DR

This paper develops a universal construction of $p$-typical Witt vectors for associative rings, extending classical concepts to non-commutative rings and relating to existing Witt functors.

Contribution

It adapts the group-theoretic universal characterization of Witt vectors to non-commutative rings, creating a universal Witt functor that generalizes classical and known non-commutative Witt constructions.

Findings

Constructed a Witt vector functor $E$ for associative rings.

Showed $E$ specializes to classical Witt vectors in the commutative case.

Introduced a universal Witt functor $ ilde{E}$ related to Hesselholt's Witt functor.

Abstract

For a prime $p$ and an associative ring $R$ with unity, there are various constructions of $p$-typical Witt vectors of $R$, all of which specialize to the classical $p$-typical Witt vectors when $R$ is commutative. These constructions are endowed with a Verschiebung operator $V$ and a Teichm\"{u}ller map $\langle \cdot \rangle$, and they satisfy the property that the map $x \mapsto V\langle x^p\rangle - p\langle x \rangle$ is additive. In this paper, we adapt the group-theoretic universal characterization of classical $p$-typical Witt vectors proposed in arXiv:2405.12680 to the non-commutative setting. Our main result is that this approach yields a construction of Witt vectors for associative rings, denoted $E$, which specializes correctly to the classical Witt functor in the commutative case. The construction of $E$ is inspired by the Witt functor of Cuntz--Deninger, and we show that $E$ is a universal pre-Witt functor, subject to an explicit conjecture concerning non-commutative polynomials. We further introduce the notion of a Witt functor and construct a universal Witt functor $\hat{E}$, which is closely related to Hesselholt's Witt functor $W_H$. We suspect that $W_H$ is, in fact, the universal Morita-invariant Witt functor.