Decomposition of the Hochschild and cyclic homology of commutative differential graded algebras

Guillermo Cortiñas,Jorge Alberto Guccione,Juan José Guccione

doi:10.1016/0022-4049(92)90045-h

Guillermo Cortiñas, Jorge Alberto Guccione + Show 1 more

Open Access

https://doi.org/10.1016/0022-4049(92)90045-h

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Abstract

We obtain an expression for the Hochschild and cyclic homology of a commutative differential graded algebra under a suitable hypothesis.

Highlights

In Theorem 2.4 of [l] the authors show that the Hochschild and cyclic homology of a free commutative differential graded k-algebra over a characteristic zero field are the corresponding homologies of a bigraded S’-chain complex which is simpler than the canonical one
This result allows them to compute the Hochschild and cyclic homology of an arbitrary commutative differential graded k-algebra (A, d) taking a free model p : (A (V), d’)- (A, d) of (A, d) and applying Theorem 2.4 of [l] to (A(V), d’). Using this technique they obtain Hodge decompositions of the Hochschild and cyclic homology of (A, d) which coincide with the ones obtained by Gerstenhaber and Schack in [3] and Loday in [5], as ViguC-Poirrier showed in [7]
In general the free models are too complex and hard to construct. With this method, it is impossible to compute the cyclic homology of a localization of the k-algebra A mentioned above

Summary

Introduction

In Theorem 2.4 of [l] the authors show that the Hochschild and cyclic homology of a free commutative differential graded k-algebra over a characteristic zero field are the corresponding homologies of a bigraded S’-chain complex which is simpler than the canonical one. This result allows them to compute the Hochschild and cyclic homology of an arbitrary commutative differential graded k-algebra (A, d) taking a free model p : (A (V), d’)- (A, d) of (A, d) and applying Theorem 2.4 of [l] to (A(V), d’) Using this technique they obtain Hodge decompositions of the Hochschild and cyclic homology of (A, d) which coincide with the ones obtained by Gerstenhaber and Schack in [3] and Loday in [5], as ViguC-Poirrier showed in [7]. At the beginning of this investigation, our purpose was precisely to solve this problem With this in mind we prove in this work that Theorem 2.4 of [l] remains valid for algebras of the form (A, @k A (V), d), with A, homologically regular over a characteristic zero field (see Definition 2.1) and V= VI CI3V, CBV, CD.

Preliminaries

The cyclic homology of a homologically regular k-CDGA

Some computations