A Hochschild-Kostant-Rosenberg theorem for cyclic homology

Abstract

Let $A$ be a commutative algebra over the field ${\mathbb F}_2 = {\mathbb Z}/2$. We show that there is a natural algebra homomorphism $\ell (A) \to HC^-_*(A)$ which is an isomorphism when $A$ is a smooth algebra. Thus, the functor $\ell$ can be viewed as an approximation of negative cyclic homology and ordinary cyclic homology $HC_*(A)$ is a natural $\ell (A)$-module. In general, there is a spectral sequence $E^2 = L_*(\ell )(A) \Rightarrow HC_*^- (A)$. We find associated approximation functors $\ell^+$ and $\ell^{per}$ for ordinary cyclic homology and periodic cyclic homology, and set up their spectral sequences. Finally, we discuss universality of the approximations.