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Cyclic Homology of Algebras

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

Abstract

There are at least three ways to construct cyclic homology from Hochschild homology. First, in his search for a non-commutative analogue of de Rham homology theory, A. Connes discovered in 1981 the following striking phenomenon:
  • the Hochschild boundary map b is still well-defined when one factors out the module AA n = A n +1 by the action of the (signed) cyclic permutation of order n + 1.

Keywords

Exact Sequence Mixed Complex Cyclic Module Cyclic Homology Hochschild Homology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliographical Comments on Chapter 2

  1. Hochschild, G., Kostant, B., Rosenberg, A., Differential forms on regular affine algebras, Trans. AMS 102 (1962), 383–408. 26#167Google Scholar
  2. Hsiang, W.-C., Staffeldt, R.E., A model for computing rational algebraic K-theory of simply connected spaces, Invent. Math. 68 (1982), 227–239. 84h: 18015MathSciNetCrossRefGoogle Scholar
  3. Connes, A., Cohomologie cyclique et foncteurs Extn, C. R. Acad. Sci. Paris Sér. A-B 296 (1983), 953–958. 86d: 18007MathSciNetGoogle Scholar
  4. Feigin, B.L., Tsygan, B.L., Cohomologie de l’algèbre de Lie des matrices de Jacobi généralisées (en russe), Funct. Anal. and Appl. 17(2) (1983), 86–87. 85c: 17008MathSciNetGoogle Scholar
  5. Jones, J.D.S., Kassel, C., Bivariant cyclic theory, K-theory 3 (1989), 339–366. 91f: 19001MathSciNetCrossRefGoogle Scholar
  6. Mccarthy, R., L’équivalence de Morita et l’homologie cyclique, C. R. Acad. Sci. Paris Sér. A-B 307 (1988), 211–215. 89k: 18028Google Scholar
  7. Karoubi, M., Homologie cyclique et K-théorie, Astérisque 149, 1987. 89c: 18019Google Scholar
  8. Andre, M., Homologie des algèbres commutatives, Grund. Math. Wissen. 206. Springer, 1974.Google Scholar
  9. Burghelea, D., Fiedorowicz, Z., Cyclic homology and algebraic K-theory of spaces II, Topology 25 (1986), 303–317. 88i: 18009bGoogle Scholar
  10. Kassel, C., Cyclic homology, comodules and mixed complexes, J. of Algebra 107 (1987), 195–216. 88k: 18019MathSciNetCrossRefGoogle Scholar
  11. Karoubi, M., Homologie cyclique et K-théorie, Astérisque 149, 1987. 89c: 18019Google Scholar
  12. Karoubi, M., Homologie cyclique des groupes et des algèbres, C. R. Acad. Sci. Paris Sér. A-B 297 (1983), 381–384. 85g: 18012MathSciNetGoogle Scholar
  13. [Qu]
    D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65–87.MathSciNetCrossRefGoogle Scholar
  14. Feigin, B.L., Tsygan, B.L., Additive K-theory and crystalline cohomology (in russian), Funkt. Anal. i. Pril. 19 (2) (1985), 52–62 88e: 18008MathSciNetGoogle Scholar
  15. Cuntz, J., Representations of quantized differential forms in non-commutative geometry. Mathematical physics, X (Leipzig, 1991), 237–251, Springer, Berlin, 1992.Google Scholar
  16. Quillen, D., Finite generation of the group K, of algebraic integers, Springer Lect. Notes in Math. 341 (1973), 179–198. 50#2305Google Scholar
  17. Quillen, D., Algebra cochains and cyclic cohomology, Publ. Math. IHES 68 (1988) 139–174. 90j: 18008MathSciNetGoogle Scholar

Copyright information

© 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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