A Hochschild-Kostant-Rosenberg theorem for cyclic homology

Let $A$ be a commutative algebra over the field ${\mathbb F}2 = {\mathbb Z}/2$\mathrm{.}Weshowthatthereisanaturalalgebrahomomorphism$\ell (A) \to HC^-(A)$whichisanisomorphismwhen$A$ is a smooth algebra. Thus, the functor $\mathrm{\ell }$ can be viewed as an approximation of negative cyclic homology and ordinary cyclic homology $H{C}_{\ast }(A)$ is a natural $\mathrm{\ell }(A)$-module. In general, there is a spectral sequence $E^2 = L_(\ell )(A) \Rightarrow HC_*^- (A)$\mathrm{.}Wefindassociatedapproximationfunctors$\ell^+$and$\ell^{per}$ for ordinary cyclic homology and periodic cyclic homology, and set up their spectral sequences. Finally, we discuss universality of the approximations.