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

Abstract

Let A A be any unital associative, possibly non-commutative ring and let p p be a prime number. Let E ( A ) E(A) be the ring of p p -typical Witt vectors as constructed by Cuntz and Deninger in [J. Algebra 440 (2015), pp. 545–593] and let W ( A ) 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 p=2 and A A is a non-commutative unital torsion free ring, then there is no surjective continuous group homomorphism from W ( A ) → H H 0 ( E ( A ) ) := E ( A ) / [ E ( A ) , E ( A ) ] ¯ 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 p and simplify the arguments used for p = 2 p=2 . We also prove that if A A a is a non-commutative unital ring, then there is no continuous map of sets H H 0 ( E ( A ) ) → W ( A ) HH_0(E(A)) \to W(A) which commutes with the ghost maps.