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) and Alain Connes (cohomology) 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, Jean-Louis Loday, 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
- HCn(A) or Hnλ(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. 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 is 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 or asymptotic and local cyclic homology due to Michael Puschnigg. The last one is very near to K-theory as it is endowed with a bivariant Chern character from KK-theory.
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 and deformation quantization of Poisson structures.
- 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