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Cyclic Homology of Algebras

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Cyclic Homology

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 301))

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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.

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Bibliographical Comments on Chapter 2

  1. Hsiang, W.-C., Staffeldt, R.E., A model for computing rational algebraic Ktheory of simply connected spaces, Invent. Math. 68 (1982), 227–239

    Article  MathSciNet  MATH  Google Scholar 

  2. Tsygan, B.L., The homology of matrix Lie algebras over rings and the Hochschild homology (en russe), Uspekhi Mat. Nauk 38 (1983), 217–218–Russ. Math. Survey 38 No. 2 (1983), 198–199

    Google Scholar 

  3. Loday, J.-L., Quillen, D., Homologie cyclique et homologie de l’algèbre de Lie des matrices, C. R. Acad. Sci. Paris Sér. A-B 296 (1983), 295–297

    MathSciNet  MATH  Google Scholar 

  4. Connes, A., Cohomologie cyclique et foncteurs Ext°, C. R. Acad. Sci. Paris Sér. A-B 296 (1983), 953–958.

    MathSciNet  MATH  Google Scholar 

  5. Mccarthy, R., L’équivalence de Morita et l’homologie cyclique, C. R. Acad. Sci. Paris Sér. A-B 307 (1988), 211–215

    MATH  Google Scholar 

  6. Karoubi, M., Homologie cyclique et K-théorie, Astérisque 149, 1987

    Google Scholar 

  7. Connes, A., Cohomologie cyclique et foncteurs Ext°, C. R. Acad. Sci. Paris Sér. A-B 296 (1983), 953–958.

    MathSciNet  MATH  Google Scholar 

  8. M. André, Homologie des algèbres commutatives, Springer Verlag, 1974.

    Google Scholar 

  9. Burghelea, D., Cyclic homology and the algebraic K-theory of spaces I, Proc. Summer Inst. on Alg. K-theory, Boulder Colorado 1983, Contemp. Math. 55, I (1986), 89–115

    Google Scholar 

  10. Kassel, C., Cyclic homology, comodules and mixed complexes, J. of Algebra 107 (1987), 195–216.

    Article  MathSciNet  MATH  Google Scholar 

  11. Karoubi, M., Homologie cyclique et régulateurs en K-théorie algébrique, C. R. Acad. Sci. Paris Sér. A-B 297 (1983), 557–560

    MathSciNet  MATH  Google Scholar 

  12. Karoubi, M., Homologie cyclique et K-théorie algébrique I et II, C. R. Acad. Sci. Paris Sér. A-B 297 (1983), 447–450 et 513–516

    Google Scholar 

  13. Karoubi, M., Homologie cyclique et K-théorie, Astérisque 149, 1987

    Google Scholar 

  14. Quillen, D., On the (co)-homology of commutative rings, Proc. Symp. Pure Math. 17 (1970), 65–87.

    Article  MathSciNet  Google Scholar 

  15. Feigin, B.L., Tsygan, B.L., Additive K-theory and crystalline cohomology (in russian), Funkt. Anal. i. Pril. 19 (2) (1985), 52–62

    Google Scholar 

  16. Quillen, D., Algebra cochains and cyclic cohomology, Publ. Math. Ihes 68 (1988) 139–174. 90j: 18008

    MathSciNet  Google Scholar 

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© 1992 Springer-Verlag Berlin Heidelberg

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Loday, JL. (1992). Cyclic Homology of Algebras. In: Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol 301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21739-9_2

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  • DOI: https://doi.org/10.1007/978-3-662-21739-9_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21741-2

  • Online ISBN: 978-3-662-21739-9

  • eBook Packages: Springer Book Archive

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