The cyclic homology of $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$

TL;DR

This paper computes the cyclic, negative cyclic, and periodic cyclic homology of a specific quotient ring over a field, building on known Hochschild homology calculations.

Contribution

It extends existing Hochschild homology results to compute various cyclic homologies for the ring $k[x_1,...,x_d]/(x_1,...,x_d)^2$ over a field.

Findings

Explicit formulas for cyclic homology groups.

Connections between Hochschild and cyclic homology in this context.

Methodology for extending Hochschild calculations to cyclic homology.

Abstract

The Hochschild homology of the ring $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ over $k$.