On the non-existence of a cyclic homology theory with coefficients

We show that one cannot construct a cyclic homology theory with coefficients that would be related to the Hochschild homology by the Connes periodicity exact sequence. We show that this is impossible even if the ideals of a given algebra have been taken as coefficients. Despite this, the cyclic homology with coefficients can be defined by restricting the class of coefficient modules. In particular, if $γ: I→A$ is a crossed module of algebras with $I$ being $H$-unital, then it is possible to define “nice” cyclic homology groups of $A$ with coefficients in $I$.

Keywords

Hochschild and cyclic homology, H-unital algebra, crossed modules of algebras

2010 Mathematics Subject Classification

16E40, 18G10, 18G50, 18G60

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Published 27 December 2012