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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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.
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.
M. Bökstedt, W.C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.
M. Bökstedt. Topological Hochschild homology. Preprint, Bielefeld, 1986.
L. Breen. Extensions du groupe additif. Inst. Hautes Études Sci. Publ. Math., 48:39–125, 1978.
B.I. Dundas. Relative K-theory and topological cyclic homology. Acta Math., 179(2):223–242, 1997.
B.I. Dundas and R. McCarthy. Stable K-theory and topological Hochschild homology. Ann. Math. (2), 140(3):685–701, 1994.
B.I. Dundas and R. McCarthy. Topological Hochschild homology of ring functors and exact categories. J. Pure Appl. Algebra, 109(3):231–294, 1996.
V. Franjou, J. Lannes, and L. Schwartz. Autour de la cohomologie de Mac Lane des corps finis. Invent. Math., 115(3):513–538, 1994.
V. Franjou and T. Pirashvili. On the Mac Lane cohomology for the ring of integers. Topology, 37(1):109–114, 1998.
T.G. Goodwillie. Cyclic homology, derivations, and the free loopspace. Topology, 24(2):187–215, 1985.
T.G. Goodwillie. Notes on the cyclotomic trace. Lecture notes for a series of seminar talks at MSRI, Spring 1990, December 1991.
A. Hattori. Rank element of a projective module. Nagoya Math. J., 25:113–120, 1965.
L. Hesselholt and I. Madsen. On the K-theory of finite algebras over Witt vectors of perfect fields. Topology, 36(1):29–101, 1997.
L. Hesselholt and I. Madsen. On the K-theory of local fields. Ann. Math. (2), 158(1):1–113, 2003.
M. Jibladze and T. Pirashvili. Cohomology of algebraic theories. J. Algebra, 137(2):253–296, 1991.
M. Karoubi and T. Lambre. Quelques classes caractéristiques en théorie des nombres. J. Reine Angew. Math., 543:169–186, 2002.
M. Larsen and A. Lindenstrauss. Topological Hochschild homology of algebras in characteristic p. J. Pure Appl. Algebra, 145(1):45–58, 2000.
M. Larsen and A. Lindenstrauss. Topological Hochschild homology and the condition of Hochschild-Kostant-Rosenberg. Commun. Algebra, 29(4):1627–1638, 2001.
A. Lindenstrauss. A relative spectral sequence for topological Hochschild homology of spectra. J. Pure Appl. Algebra, 148(1):77–88, 2000.
A. Lindenstrauss and I. Madsen. Topological Hochschild homology of number rings. Trans. Am. Math. Soc., 352(5):2179–2204, 2000.
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.
S. Mac Lane. Homologie des anneaux et des modules. In Colloque de Topologie Algébrique, Louvain, 1956, pages 55–80. Georges Thone, Liège, 1957.
I. Madsen. Algebraic K-theory and traces. In Current Developments in Mathematics, Cambridge, MA, 1995, pages 191–321. International Press, Cambridge, 1994.
R. McCarthy. The cyclic homology of an exact category. J. Pure Appl. Algebra, 93(3):251–296, 1994.
T. Pirashvili. On the topological Hochschild homology of Z/p k Z. Commun. Algebra, 23(4):1545–1549, 1995.
T. Pirashvili. Polynomial approximation of Ext and Tor groups in functor categories. Commun. Algebra, 21(5):1705–1719, 1993.
T. Pirashvili. Spectral sequence for Mac Lane homology. J. Algebra, 170(2):422–428, 1994.
T. Pirashvili and F. Waldhausen. Mac Lane homology and topological Hochschild homology. J. Pure Appl. Algebra, 82(1):81–98, 1992.
C. Schlichtkrull. The transfer map in topological Hochschild homology. J. Pure Appl. Algebra, 133(3):289–316, 1998.
R. Schwänzl, R.M. Vogt, and F. Waldhausen. Topological Hochschild homology. J. Lond. Math. Soc. (2), 62(2):345–356, 2000.
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.
G. Segal. Categories and cohomology theories. Topology, 13:293–312, 1974.
U. Shukla. A relative cohomology for associative algebras. Proc. Am. Math. Soc., 15:461–469, 1964.
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.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag London
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4393-2_4
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4392-5
Online ISBN: 978-1-4471-4393-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)