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Part of the book series: Algebra and Applications ((AA,volume 18))

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Abstract

Topological Hochschild homology is introduced along with its natural extension to enriched model categories. Several basic properties such as Morita invariance and invariance under weak equivalences of enriched categories are demonstrated.

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References

  1. A.J. Berrick and L. Hesselholt. Topological Hochschild homology and the Bass trace conjecture. Available from http://www-math.mit.edu/~larsh/papers/029/, 2008.

  2. M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W.C. Hsiang, and I. Madsen. On the algebraic K-theory of simply connected spaces. Duke Math. J., 84(3):541–563, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  3. M. Bökstedt, W.C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bökstedt. Topological Hochschild homology. Preprint, Bielefeld, 1986.

    Google Scholar 

  5. L. Breen. Extensions du groupe additif. Inst. Hautes Études Sci. Publ. Math., 48:39–125, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  6. B.I. Dundas. Relative K-theory and topological cyclic homology. Acta Math., 179(2):223–242, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  7. B.I. Dundas and R. McCarthy. Stable K-theory and topological Hochschild homology. Ann. Math. (2), 140(3):685–701, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  8. B.I. Dundas and R. McCarthy. Topological Hochschild homology of ring functors and exact categories. J. Pure Appl. Algebra, 109(3):231–294, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  9. V. Franjou, J. Lannes, and L. Schwartz. Autour de la cohomologie de Mac Lane des corps finis. Invent. Math., 115(3):513–538, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Franjou and T. Pirashvili. On the Mac Lane cohomology for the ring of integers. Topology, 37(1):109–114, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  11. T.G. Goodwillie. Cyclic homology, derivations, and the free loopspace. Topology, 24(2):187–215, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  12. T.G. Goodwillie. Notes on the cyclotomic trace. Lecture notes for a series of seminar talks at MSRI, Spring 1990, December 1991.

    Google Scholar 

  13. A. Hattori. Rank element of a projective module. Nagoya Math. J., 25:113–120, 1965.

    MathSciNet  MATH  Google Scholar 

  14. L. Hesselholt and I. Madsen. On the K-theory of finite algebras over Witt vectors of perfect fields. Topology, 36(1):29–101, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Hesselholt and I. Madsen. On the K-theory of local fields. Ann. Math. (2), 158(1):1–113, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. Jibladze and T. Pirashvili. Cohomology of algebraic theories. J. Algebra, 137(2):253–296, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Karoubi and T. Lambre. Quelques classes caractéristiques en théorie des nombres. J. Reine Angew. Math., 543:169–186, 2002.

    MathSciNet  MATH  Google Scholar 

  18. M. Larsen and A. Lindenstrauss. Topological Hochschild homology of algebras in characteristic p. J. Pure Appl. Algebra, 145(1):45–58, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Larsen and A. Lindenstrauss. Topological Hochschild homology and the condition of Hochschild-Kostant-Rosenberg. Commun. Algebra, 29(4):1627–1638, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Lindenstrauss. A relative spectral sequence for topological Hochschild homology of spectra. J. Pure Appl. Algebra, 148(1):77–88, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Lindenstrauss and I. Madsen. Topological Hochschild homology of number rings. Trans. Am. Math. Soc., 352(5):2179–2204, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  22. J.-L. Loday. Cyclic Homology, 2nd edition, volume 301 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1998. Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili.

    MATH  Google Scholar 

  23. S. Mac Lane. Homologie des anneaux et des modules. In Colloque de Topologie Algébrique, Louvain, 1956, pages 55–80. Georges Thone, Liège, 1957.

    Google Scholar 

  24. I. Madsen. Algebraic K-theory and traces. In Current Developments in Mathematics, Cambridge, MA, 1995, pages 191–321. International Press, Cambridge, 1994.

    Google Scholar 

  25. R. McCarthy. The cyclic homology of an exact category. J. Pure Appl. Algebra, 93(3):251–296, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  26. T. Pirashvili. On the topological Hochschild homology of Z/p k Z. Commun. Algebra, 23(4):1545–1549, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  27. T. Pirashvili. Polynomial approximation of Ext and Tor groups in functor categories. Commun. Algebra, 21(5):1705–1719, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  28. T. Pirashvili. Spectral sequence for Mac Lane homology. J. Algebra, 170(2):422–428, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Pirashvili and F. Waldhausen. Mac Lane homology and topological Hochschild homology. J. Pure Appl. Algebra, 82(1):81–98, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  30. C. Schlichtkrull. The transfer map in topological Hochschild homology. J. Pure Appl. Algebra, 133(3):289–316, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Schwänzl, R.M. Vogt, and F. Waldhausen. Topological Hochschild homology. J. Lond. Math. Soc. (2), 62(2):345–356, 2000.

    Article  MATH  Google Scholar 

  32. R. Schwänzl, R. Staffeldt, and F. Waldhausen. Stable K-theory and topological Hochschild homology of A rings. In Algebraic K-Theory, Poznań, 1995, volume 199 of Contemporary Mathematics, pages 161–173. Amer. Math. Soc., Providence, 1996.

    Google Scholar 

  33. G. Segal. Categories and cohomology theories. Topology, 13:293–312, 1974.

    Article  MathSciNet  MATH  Google Scholar 

  34. U. Shukla. A relative cohomology for associative algebras. Proc. Am. Math. Soc., 15:461–469, 1964.

    Article  MATH  Google Scholar 

  35. J.R. Stallings. On infinite processes leading to differentiability in the complement of a point. In Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pages 245–254. Princeton University Press, Princeton, 1965.

    Google Scholar 

  36. F. Waldhausen. Algebraic K-theory of topological spaces, II. In Algebraic Topology, Proc. Sympos., Univ. Aarhus, Aarhus, 1978, volume 763 of Lecture Notes in Mathematics, pages 356–394. Springer, Berlin, 1979.

    Google Scholar 

  37. F. Waldhausen. Algebraic K-theory of spaces, concordance, and stable homotopy theory. In Algebraic Topology and Algebraic K-Theory, Princeton, NJ, 1983, volume 113 of Annals of Mathematics Studies, pages 392–417. Princeton University Press, Princeton, 1987.

    Google Scholar 

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Dundas, B.I., Goodwillie, T.G., McCarthy, R. (2013). Topological Hochschild Homology. In: The Local Structure of Algebraic K-Theory. Algebra and Applications, vol 18. Springer, London. https://doi.org/10.1007/978-1-4471-4393-2_4

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