The Dennis Supertrace and the Hochschild Homology of Supermatrices

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.

💡 Research Summary

The paper develops a super‑version of the Dennis trace, a classical map that relates the Hochschild homology of a ring R to that of the matrix algebra Mₙ(R). The authors work with an arbitrary (not necessarily commutative) super‑algebra A equipped with a unit and consider the (p|q)‑type supermatrix algebra Mₚ|₍q₎(A). After recalling the necessary background on super‑algebras, super‑matrices, and Hochschild complexes, they introduce the super‑trace str: Mₚ|₍q₎(A) → A, defined by str(M) = tr(X) – tr(W) for a block matrix M =