# Cyclic homology

In noncommutative geometry and related branches of mathematics, **cyclic homology** and **cyclic cohomology** are certain (co)homology theories for associative algebras which generalize the de Rham (co)homology of manifolds. These notions were independently introduced by Boris Tsygan (homology)^{[1]} and Alain Connes (cohomology)^{[2]} in 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the K-theory. The principal contributors to the development of theory include Max Karoubi, Yuri L. Daletskii, Boris Feigin, Jean-Luc Brylinski, Mariusz Wodzicki, Template:Link-interwiki, Victor Nistor, Daniel Quillen, Joachim Cuntz, Ryszard Nest, Ralf Meyer, Michael Puschnigg, and many others.

## Hints about definition

The first definition of the cyclic homology of a ring *A* over a field of characteristic zero, denoted

*HC*_{n}(*A*) or*H*_{n}^{λ}(*A*),

proceeded by the means of an explicit chain complex related to the Hochschild homology complex of *A*. Connes later found a more categorical approach to cyclic homology using a notion of **cyclic object** in an abelian category, which is analogous to the notion of simplicial object. In this way, cyclic homology (and cohomology) may be interpreted as a derived functor, which can be explicitly computed by the means of the (*b*, *B*)-bicomplex.

One of the striking features of cyclic homology is the existence of a long exact sequence connecting Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.

## Case of commutative rings

Cyclic cohomology of the commutative algebra *A* of regular functions on an affine algebraic variety over a field *k* of characteristic zero can be computed in terms of Grothendieck's algebraic de Rham complex.^{[3]} In particular, if the variety *V*=Spec *A* is smooth, cyclic cohomology of *A* are expressed in terms of the de Rham cohomology of *V* as follows:

This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra *A*, which was extensively developed by Connes.

## Variants of cyclic homology

One motivation of cyclic homology was the need for an approximation of K-theory that be defined, unlike K-theory, as the homology of a chain complex. Cyclic cohomology is in fact endowed with a pairing with K-theory, and one hopes this pairing to be non-degenerate.

There has been defined a number of variants whose purpose is to fit better with algebras with topology, such as Fréchet algebras, -algebras, etc. The reason is that K-theory behaves much better on topological algebras such as Banach algebras or C*-algebras than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories. Among them are entire cyclic homology due to Alain Connes, analytic cyclic homology due to Ralf Meyer^{[4]} or asymptotic and local cyclic homology due to Michael Puschnigg.^{[5]} The last one is very near to K-theory as it is endowed with a bivariant Chern character from KK-theory.

## Applications

One of the applications of cyclic homology is to find new proofs and generalizations of the Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triples^{[6]} and deformation quantization of Poisson structures.^{[7]}

## See also

## References

- ↑ Boris L. Tsygan. Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk, 38(2(230)):217–218, 1983. Translation in Russ. Math. Survey 38(2) (1983), 198–199.
- ↑ Alain Connes. Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math., 62:257–360, 1985.
- ↑ Boris L. Fegin and Boris L. Tsygan. Additive K-theory and crystalline cohomology. Funktsional. Anal. i Prilozhen., 19(2):52–62, 96, 1985.
- ↑ Ralf Meyer. Analytic cyclic cohomology. PhD thesis, Universität Münster, 1999
- ↑ Michael Puschnigg. Diffeotopy functors of ind-algebras and local cyclic cohomology. Doc. Math., 8:143–245 (electronic), 2003.
- ↑ Alain Connes and Henri Moscovici. The local index formula in noncommutative geometry. Geom. Funct. Anal., 5(2):174–243, 1995.
- ↑ Ryszard Nest and Boris Tsygan. Algebraic index theorem. Comm. Math. Phys., 172(2):223–262, 1995.

- Jean-Louis Loday,
*Cyclic Homology*, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0