Cyclic homology of commutative algebras over general ground rings

TL;DR

This paper develops spectral sequences to compute cyclic and Hochschild homology of commutative algebras over general ground rings, extending known results and providing new formulas especially for complete intersections.

Contribution

It introduces spectral sequences for cyclic and Hochschild homology over arbitrary ground rings and establishes conditions for their degeneration, generalizing previous characteristic zero results.

Findings

Spectral sequences for cyclic and Hochschild homology are constructed.

Degeneration of spectral sequences is proven under specific conditions.

Formulas for Hochschild homology of complete intersections are derived.

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).