Cyclic homology of algebras of global dimension at most two

TL;DR

This paper investigates the cyclic homology of certain graded algebras, providing explicit formulas and conditions under which the homology vanishes, especially for algebras of global dimension two.

Contribution

It introduces a new approach to compute cyclic homology for algebras of global dimension two, including explicit formulas and a refined notion of 'strongly free' relations.

Findings

Cyclic homology vanishes in degrees greater than one for global dimension two algebras.

For monomial relations, cyclic homology also vanishes in degree one.

Explicit formulas are provided for tensor algebras modulo symmetric quadratic forms.

Abstract

We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to prove that cyclic homology of an algebra of global dimension two is zero in homological degree greater than one and is zero also in homological degree equal to one in case the relations are monomials. We give also explicit formulas for the cyclic homology of a tensor algebra modulo one symmetric quadratic form.