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.
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
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.
Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories.
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 a class in K homology.
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.
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
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.