A Hochschild-Kostant-Rosenberg theorem for cyclic homology

Let A be a commutative algebra over the field F2=Z/2. We show that there is a natural algebra homomorphism ℓ(A)→HC⁎−(A) which is an isomorphism when A is a smooth algebra. Thus, the functor ℓ can be viewed as an approximation of negative cyclic homology and ordinary cyclic homology HC⁎(A) is a natural ℓ(A)-module. In general, there is a spectral sequence E2=L⁎(ℓ)(A)⇒HC⁎−(A). We find associated approximation functors ℓ+ and ℓper for ordinary cyclic homology and periodic cyclic homology, and set up their spectral sequences. Finally, we discuss universality of the approximations.