Homology, Homotopy and Applications

Volume 20 (2018)

Number 1

Cyclic vs mixed homology

Pages: 237 – 250


Ulrich Krähmer (Institut für Geometrie, Technische Universität, Dresden, Germany)

Dylan Madden (Mathematics Institute, University of Warwick, Coventry, United Kingdom)


The spectral theory of the Karoubi operator due to Cuntz and Quillen is extended to general mixed (duchain) complexes, that is, chain complexes which are also cochain complexes. Connes’ coboundary map $B$ can be viewed as a perturbation of the noncommutative De Rham differential by a polynomial in the Karoubi operator. The homological impact of such perturbations is expressed in terms of two short exact sequences.


mixed complex, cyclic homology, mixed homology, noncommutative differential form

2010 Mathematics Subject Classification

18G30, 18G35, 18G60, 19D55

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U.K. thanks Gabriella Böhm, Niels Kowalzig and Tomasz Maszczyk for discussions, and IMPAN Warsaw and the Wigner Institute Budapest for hospitality. Both authors thank the referee for a very careful reading of the manuscript that led to numerous improvements.

Received 18 May 2017

Received revised 18 August 2017

Published 14 February 2018