A new construction of cyclic homology

TL;DR

This paper introduces a simplified construction of cyclic homology for unital algebras using noncommutative de Rham complexes, and connects it to equivariant Deligne cohomology through extended cyclic homology.

Contribution

It provides a new, streamlined approach to cyclic homology based on noncommutative de Rham complexes and establishes a novel link to equivariant Deligne cohomology.

Findings

Constructed a simple model of cyclic homology using noncommutative de Rham complexes.

Defined and computed extended cyclic homology based on an extended noncommutative de Rham complex.

Established a natural map from cyclic homology to equivariant Deligne cohomology for algebra representations.

Abstract

Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.