Morphisms between two constructions of Witt vectors of non-commutative rings

Abstract

Let $A$ be any unital associative, possibly non-commutative ring 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 the abelian group constructed by Hesselholt. In arXiv:1708.04065 it was proved that if $ p=2$ and $A$ is non commutative unital torsion free ring then there is no surjective continuous group homomorphism from $W(A) \to HH_0(E(A)): = E(A)/\overline{[E(A),E(A)]}$ which commutes with the Verschiebung operator and the Teichm\"uller map. In this paper we generalise this result to all primes $p$ and simplify the arguments used for $p=2$. We also prove that if $A$ a is non-commutative unital ring then there is no continuous map of sets $HH_0(E(A)) \to W(A)$ which commutes with the ghost maps.