Variations on Cyclic Homology
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.
KeywordsExact Sequence Cyclic Module Topological Algebra Cyclic Homology Hochschild Homology
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Bibliographical Comments on Chapter 5
- 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 , 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  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 , 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.Dihedral and quaternionic homology were introduced and studied in Loday . Independent and similar work appeared in Krasauskas-Lapin-Solovev  and Krasauskas-Solovev [1986, 1988 ]. Subsequent work was done in Lodder [1990, 1992] and in Dunn  where the relationship with 0(2)-spaces is also worked out. An interesting application to higher Arf invariants is done in Wolters .Google Scholar
- 3-4. The extension of HC to DG-algebras appeared in Vigué-Burghelea  and also Goodwillie [1985a]. The idea of getting a decomposition of HC from this point of view is in Burghelea-Vigué . Extensive computations have been made in loc. cit., Brylinski [1987b], Vigué [1988, 1990), Geller-Reid-Weibel , Bach , Hanlon [ 1986 ]. Some of these results can be found in Feigin-Tsygan [FT].Google Scholar
- 5.Bivariant cyclic cohomology was taken out from Jones-Kassel , see also Kassel [1989a]. The A-decomposition is in Nuss .Google Scholar
- 6.Some computations in the topological framework are done in Connes [C]. En-tire cyclic cohomology is treated in Connes  and used extensively for the proof of some cases of the Novikov conjecture in Connes-Moscovici  and Connes-Gromov-Moscovici [19901). Further work can be found in Connes-Gromov-Moscovici ) (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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