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 $\ell (A)\to H{C}_{\ast }^{-}(A)$ 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 $H{C}_{\ast }(A)$ is a natural $\ell (A)$-module. In general, there is a spectral sequence $E^2=L_{\ast }(\ell )(A)\Rightarrow H{C}_{\ast }^{-}(A)$. 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.

To appear in J. Pure Appl. Algebra