On the comparison of two constructions of Witt vectors of non-commutative rings

TL;DR

This paper compares two constructions of Witt vectors for non-commutative rings and demonstrates that, for p=2, they are not isomorphic if certain operators are preserved.

Contribution

It provides a negative answer to Hesselholt's question about the isomorphism of two Witt vector constructions for non-commutative rings at p=2.

Findings

No isomorphism exists between the two constructions at p=2 when preserving key operators.

The result clarifies limitations in relating different Witt vector theories for non-commutative rings.

The paper highlights the importance of operator compatibility in Witt vector isomorphisms.

Abstract

Let $A$ be any associative ring , possibly non-commutative, and let $p$ be a prime number. Let $E(A)$ be the ring of $p$-typical Witt vectors as constructed by Cuntz and Deninger and $W(A)$ be that constructed by Hesselholt. The goal of this paper is to answer the following question by Hesselholt: Is $HH_0(E(A)) $ isomorphic to $W(A)$? We show that in the case $p=2$, there is no such isomorphism possible if one insists it to be compatible with the Verscheibung operator and the Teichm\"uller map.