Periodicity and Cyclic Homology. Para-S-Modules and Perturbation Lemmas

TL;DR

This paper introduces para-$S$-modules, a new framework extending cyclic modules, and develops perturbation lemmas to compare and analyze their homological properties, with applications to periodic cyclic homology.

Contribution

It defines para-$S$-modules, generalizes perturbation lemmas for them, and establishes comparison results with classical cyclic complexes, advancing the understanding of periodic cyclic homology.

Findings

Established comparison results between para-$S$-modules and cyclic complexes.

Provided explicit methods to convert periodic cocycles.

Extended perturbation lemmas to para-$S$-modules.

Abstract

In this paper, we introduce a paracyclic version of $S$-modules. These new objects are called para-$S$-modules. Paracyclic modules and parachain complexes give rise to para-$S$-modules much in the same way as cyclic modules and mixed complexes give rise to $S$-modules. More generally, para-$S$-modules provide us with a natural framework to get analogues for paracyclic modules and parachain complexes of various constructions and equivalence results for cyclic modules or mixed complexes. The datum of a para-$S$-module does not provide us with a chain complex, and so notions of homology and quasi-isomorphisms do not make sense. We establish some generalizations for para-$S$-modules and parachain complexes of the basic perturbation lemma of differential homological algebra. These generalizations provide us with general recipes for converting deformation retracts of Hoschschild chain complexes into deformation retracts of para-$S$-modules. By using ideas of Kassel this then allows us to get comparison results between the various para-$S$-modules associated with para-precyclic modules, and between them and Connes' cyclic chain complex. These comparison results lead us to alternative descriptions of Connes' periodicity operator. This has some applications in periodic cyclic homology. We also describe the counterparts of these results in cyclic cohomology. In particular, we obtain an explicit way to convert a periodic $(b,B)$-cocycle into a cohomologous periodic cyclic cocycle.