Cyclic homology of cleft extensions of algebras

TL;DR

This paper introduces a simplified mixed complex for computing Hochschild and cyclic homologies of cleft algebra extensions, featuring a harmonic decomposition similar to known algebraic structures.

Contribution

It presents a new, simpler mixed complex for Hochschild and cyclic homology of cleft extensions, with harmonic decomposition properties.

Findings

The complex effectively computes homologies relative to ker(p).

It exhibits a harmonic decomposition akin to Cuntz and Quillen's work.

The approach simplifies previous methods for analyzing such algebra extensions.

Abstract

Let k be a commutative algebra with the field of the rational numbers included in k and let (E,p,i) be a cleft extension of A. We obtain a new mixed complex, simpler than the canonical one, giving the Hochschild and cyclic homologies of E relative to ker(p). This complex resembles the canonical reduced mixed complex of an augmented algebra. We begin the study of our complex showing that it has a harmonic decomposition like to the one considered by Cuntz and Quillen for the normalized mixed complex of an algebra.