Cyclic homology of commutative algebras over general ground rings

Abstract

We consider commutative algebras and chain DG algebras over a fixed commutative ground ring $k$ as in the title. We are concerned with the problem of computing the cyclic (and Hochschild) homology of such algebras via free DG-resolutions $\Lambda V @>>> A$. We find spectral sequences $$E^2_{p,q}=H_p(\Lambda V\otimes\Gamma^q(dV))\Rightarrow HH_{p+q}(\Lambda V)$$ and $${E'}^2_{\pq}=H_p(\Lambda V\otimes\Gamma^{\le q}(dV)) \Rightarrow HC_{p+q}(\Lambda V)$$ The algebra $\Lambda V\otimes\Gamma(dV)$ is a divided power version of the de Rham algebra; in the particular case when $k$ is a field of characteristic zero, the spectral sequences above agree with those found by Burghelea and Vigu\'e (Cyclic homology of commutative algebras I, Lecture Notes in Math. {\bf 1318} (1988) 51-72), where it is shown they degenerate at the $E^2$ term. For arbitrary ground rings we prove here (Theorem 2.3) that if $V_n=0$ for $n\ge 2$ then $E^2=E^\infty$. From this we derive a formula for the Hochschild homology of flat complete intersections in terms of a filtration of the complex for crystalline cohomology, and find a description of ${E'}^2$ also in terms of crystalline cohomology (theorem 3.0). The latter spectral sequence degenerates for complete intersections of embedding dimension $\le 2$ (Corollary 3.1). Without flatness assumptions, our results can be viewed as the computation Shukla (cyclic) homology (T. Pirashvili, F. Waldhausen; Mac Lane homology and topological Hochschild homology, J. Pure Appl. Algebra{\bf 82} (1992) 81-98).