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

TL;DR

The paper develops a universal, noncommutative analogue of $p$-typical Witt vectors for associative rings by constructing a functor $E:{\mathsf{Rings}}\to{\mathsf{Ab}}$ inspired by the Cuntz–Deninger approach. It proves that $E$ is a pre-Witt functor extending the classical Witt vectors on commutative rings, and it introduces a universal Witt functor $\hat{E}$ via quotient by Witt-relations, relating it to Hesselholt's $W_H$ and exploring Morita-invariance questions. A central conjecture (Conjecture $\ref{['zind']}$) on noncommutative polynomials is key to achieving universality; under this conjecture, $E$ becomes universal as a pre-Witt functor and $\hat{E}$ attains a universal Witt functor status with a natural map to $W_H$. The work also provides a thorough framework for noncommutative Witt polynomials, defines a noncommutative analogue of Witt sum and difference, and offers computational evidence supporting the conjecture, laying groundwork for Morita-invariant Witt theories. Overall, this establishes a robust foundation for noncommutative Witt vectors and their connections to existing Witt theories.

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 Teichmüller 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.

Theorems & Definitions (45)

• Theorem 1.1
• Definition 1.2: pre-Witt functor
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.7
• Conjecture 1.9
• Definition 2.1: $X(R)$
• Definition 2.2: $X_I$ and $X_I^{{\rm sat}}$
• Remark 2.3