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 the 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. Contributors to the development of the 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, and Michael Puschnigg.

## 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 the following explicit chain complex related to the Hochschild homology complex of *A*, called the **Connes complex**:

For any natural number *n ≥ 0*, define the operator which generates the natural cyclic action of on the *n*-th tensor product of *A*:

Recall that the Hochschild complex groups of *A* with coefficients in *A* itself are given by setting for all *n ≥ 0*. Then the components of the Connes complex are defined as , and the differential is the restriction of the Hochschild differential to this quotient. One can check that the Hochschild differential does indeed factor through to this space of coinvariants.
^{[3]}

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. If the field *k* contains the rational numbers, the definition in terms of the Connes complex calculates the same homology.

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.^{[4]} 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^{[5]} or asymptotic and local cyclic homology due to Michael Puschnigg.^{[6]} The last one is very close 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^{[7]} and deformation quantization of Poisson structures.^{[8]}

An elliptic operator D on a compact smooth manifold defines an class in K homology. One invariant of this class is the analytic index of the operator. This is seen as the pairing of the class [D], with the element 1 in HC(C(M)). Cyclic cohomology can be seen as a way to get higher invariants of elliptic differential operators not only for smooth manifolds, but also for foliations, orbifolds, and singular spaces that appear in noncommutative geometry.

## Computations of algebraic K-theory

The cyclotomic trace map is a map from algebraic K-theory (of a ring *A*, say), to cyclic homology:

In some situations, this map can be used to compute K-theory by means of this map. A pioneering result in this direction is a theorem of Goodwillie (1986): it asserts that the map

between the relative K-theory of *A* with respect to a *nilpotent* two-sided ideal *I* to the relative cyclic homology (measuring the difference between K-theory or cyclic homology of *A* and of *A*/*I*) is an isomorphism for *n*≥1.

While Goodwillie's result holds for arbitrary rings, a quick reduction shows that it is in essence only a statement about . For rings not containing **Q**, cyclic homology must be replaced by topological cyclic homology in order to keep a close connection to K-theory. (If **Q** is contained in *A*, then cyclic homology and topological cyclic homology of *A* agree.) This is in line with the fact that (classical) Hochschild homology is less well-behaved than topological Hochschild homology for rings not containing **Q**. Clausen, Mathew & Morrow (2018) proved a far-reaching generalization of Goodwillie's result, stating that for a commutative ring *A* so that the Henselian lemma holds with respect to the ideal *I*, the relative K-theory is isomorphic to relative topological cyclic homology (without tensoring both with **Q**). Their result also encompasses a theorem of Gabber (1992), asserting that in this situation the relative K-theory spectrum modulo an integer *n* which is invertible in *A* vanishes. Jardine (1993) used Gabber's result and Suslin rigidity to reprove Quillen's computation of the K-theory of finite fields.

## See also

## Notes

**^**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.**^**Jean-Louis Loday. Cyclic Homology. Vol. 301. Springer Science & Business Media, 1997.**^**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.

## References

- Jardine, J. F. (1993), "The K-theory of finite fields, revisited",
*K-Theory*,**7**(6): 579–595, doi:10.1007/BF00961219, MR 1268594 - Loday, Jean-Louis (1998),
*Cyclic Homology*, Grundlehren der mathematischen Wissenschaften,**301**, Springer, ISBN 978-3-540-63074-6 - Gabber, Ofer (1992), "
*K*-theory of Henselian local rings and Henselian pairs",*Algebraic*K*-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989)*, Contemp. Math.,**126**, AMS, pp. 59–70 - Clausen, Dustin; Mathew, Akhil; Morrow, Matthew (2018), "K-theory and topological cyclic homology of henselian pairs", arXiv:1803.10897 [math.KT]
- Goodwillie, Thomas G. (1986), "Relative algebraic
*K*-theory and cyclic homology",*Annals of Mathematics*, Second Series,**124**(2): 347–402, doi:10.2307/1971283, JSTOR 1971283, MR 0855300 - Rosenberg, Jonathan (1994),
*Algebraic K-theory and its applications*, Graduate Texts in Mathematics,**147**, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94248-3, MR 1282290, Zbl 0801.19001 CS1 maint: discouraged parameter (link). Errata