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


Author: Supriya Pisolkar
Journal: Proc. Amer. Math. Soc. 148 (2020), 2835-2842
MSC (2010): Primary 16S99; Secondary 13F35
DOI: https://doi.org/10.1090/proc/14992
Published electronically: March 30, 2020
MathSciNet review: 4099773
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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.


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Additional Information

Supriya Pisolkar
Affiliation: Indian Institute of Science, Education and Research (IISER), Homi Bhabha Road, Pashan, Pune - 411008, India
MR Author ID: 868359
Email: supriya@iiserpune.ac.in

Received by editor(s): April 28, 2019
Received by editor(s) in revised form: November 13, 2019
Published electronically: March 30, 2020
Additional Notes: This work was supported by the SERB-MATRICS grant MTR/2018/000346.
Communicated by: Sarah Witherspoon
Article copyright: © Copyright 2020 American Mathematical Society