Abstract
This chapter is devoted to the computation of Hochschild and cyclic homologies of some particular types of algebras: tensor algebras, symmetric algebras, universal enveloping algebras of Lie algebras and, finally, smooth algebras, on which we put some emphasis. One more important example, the case of group algebras, will be treated later, in Sect. 7.4. It is also shown in this chapter that Hochschild and cyclic homology are related to many other theories such as the homology of Lie algebras, André-Quillen homology of commutative algebras, and Deligne cohomology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliographical Comments on Chapter 3
D. Quillen, On the (co)-homology of commutative rings, Proc. Symp. Pure Math. 17 (1970) 65–87.
Kassel, C., L’homologie cyclique des algèbres enveloppantes, Invent. Math. 91 (1988), 221–251. 89e: 17015
Feigin, B.L., Tsygan, B.L., Additive K-theory, in “K-theory, Arithmetic and Geometry”, Springer Lect. Notes in Math. 1289 (1987), 97–209. 89a: 18017
Baum, P., Connes, A., Chern character for discrete groups, in Fête of topology, Academic Press (1988), 163–232. 90e: 58149
Cartan, H., Eilenberg, S., Homological algebra, Princeton University Press, 1956.
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: 18015
M. André, Homologie des algèbres commutatives, Springer Verlag, 1974.
Harrison, D.K., Commutative algebras and cohomology, Trans. AMS 104 (1962), 191–204.
Avramov, L., Halperin, S., On the vanishing of cotangent cohomology, Com-ment. Math. Helv. 62 (1987), 169–184.
Gros, M., Quelques résultats sur l’homologie cyclique des algèbres en caractéristique positive, C. R. Acad. Sci. Paris Sér. A-B 304 (1987), 139–142. 88i: 18015
Wodzicki, M., Cyclic homology of pseudodifferential operators and noncommutative Euler class, C. R. Acad. Sci. Paris Sér. A-B 306 (1988), 321–325. 89h: 58189
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Loday, JL. (1998). Smooth Algebras and Other Examples. In: Cyclic Homology. Grundlehren der mathematischen Wissenschaften, vol 301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-11389-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-662-11389-9_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-08316-7
Online ISBN: 978-3-662-11389-9
eBook Packages: Springer Book Archive