Variations on Cyclic Homology

  • Jean-Louis Loday
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 301)


There are several ways of modifying cyclic homology: by altering the cyclic bicomplex, by putting up other groups than the cyclic groups or by enlarging the category of algebras.


Exact Sequence Cyclic Module Topological Algebra Cyclic Homology Hochschild Homology 
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Bibliographical Comments on Chapter 5

  1. 1.
    The idea of looking at the periodic complex, and so at the periodic theory, is already in the seminal article of Connes [C], see also Goodwillie [1985a]. The idea that the theory HC- is relevant is due to Hood-Jones [1987], where they recognize this theory as the dual of the cyclic theory over Ha. (k). Similar statements can be found in Feigin-Tsygan [FT]. The product structure on HC- was introduced in Hood-Jones [1987] by using the acyclic model technique. De Rham cohomology has been generalized to crystalline cohomology by Grothendieck and the comparison with the periodic cyclic theory is done in Feigin-Tsygan [1987], see also Kassel [1987, cor. 3.12]. In the literature periodic cyclic homology is denoted either by HCP (adopted here), or PHC,or HCP,or HP,or even simply H. Google Scholar
  2. 2.
    Dihedral and quaternionic homology were introduced and studied in Loday [1987]. Independent and similar work appeared in Krasauskas-Lapin-Solovev [1987] and Krasauskas-Solovev [1986, 1988 ]. Subsequent work was done in Lodder [1990, 1992] and in Dunn [1989] where the relationship with 0(2)-spaces is also worked out. An interesting application to higher Arf invariants is done in Wolters [1992].Google Scholar
  3. 3-4. The extension of HC to DG-algebras appeared in Vigué-Burghelea [1985] and also Goodwillie [1985a]. The idea of getting a decomposition of HC from this point of view is in Burghelea-Vigué [1988]. Extensive computations have been made in loc. cit., Brylinski [1987b], Vigué [1988, 1990), Geller-Reid-Weibel [1989], Bach [1992], Hanlon [ 1986 ]. Some of these results can be found in Feigin-Tsygan [FT].Google Scholar
  4. 5.
    Bivariant cyclic cohomology was taken out from Jones-Kassel [1989], see also Kassel [1989a]. The A-decomposition is in Nuss [1992].Google Scholar
  5. 6.
    Some computations in the topological framework are done in Connes [C]. En-tire cyclic cohomology is treated in Connes [1988] and used extensively for the proof of some cases of the Novikov conjecture in Connes-Moscovici [1990] and Connes-Gromov-Moscovici [19901). Further work can be found in Connes-Gromov-Moscovici [1992]) (asymptotic cyclic cohomology, again in relationship with the Novikov con-jecture). Many other papers relating the index theory and the entire cyclic coho-mology are listed in the references.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jean-Louis Loday
    • 1
  1. 1.Centre National de la Recherche ScientifiqueInstitut de Recherche Mathématique AvancéeStrasbourgFrance

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