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 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 and deformation quantization of Poisson structures. An elliptic operator D on a compact smooth manifold defines a 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 : 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