The Dennis' Supertrace and the Hochschild Homology of Supermatrices

TL;DR

This paper generalizes the Dennis trace to supermatrices over superalgebras and demonstrates an isomorphism between their Hochschild homologies, extending classical results to the superalgebra context.

Contribution

It introduces a supermatrix Dennis trace and proves an isomorphism between Hochschild homologies of superalgebras and supermatrix algebras.

Findings

Established a supermatrix Dennis trace construction.

Proved Hochschild homology isomorphism for superalgebras.

Extended classical Hochschild homology results to superalgebras.

Abstract

We construct, in this paper, a generalization of the Dennis trace (for matrices) to the case of the supermatrices over an arbitrary (not necessarily commutative) superalgebra with unit. By analogy with the ungraded case, we show how it is possible to use this map to construct an isomorphism from the Hochschild homology of the superalgebra to the Hochschild homology of the supermatrix algebra.