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 in [J. Algebra 440 (2015), pp. 545–593] and let $W(A)$ be the abelian group constructed by Hesselholt in [Acta Math. 178 (1997), pp. 109–141] and [Acta Math. 195 (2005), pp. 55–60]. In [J. Algebra 506 (2018), pp. 379–396] it was proved that if $p=2$ and $A$ is a 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 Teichmüller 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 a 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.