Hochschild homology invariants of K\"ulshammer type of derived categories

TL;DR

This paper extends the concept of K"ulshammer invariants from symmetric to non-symmetric finite-dimensional algebras by developing a Hochschild homology-based framework applicable beyond symmetric cases.

Contribution

It generalizes K"ulshammer invariants to higher Hochschild homology for non-symmetric algebras using a novel approach involving trivial extension algebras.

Findings

Established a Hochschild homology analogue of K"ulshammer invariants for non-symmetric algebras.

Extended the duality of K"ulshammer ideals to higher Hochschild homology.

Provided a new framework for invariants in derived categories.

Abstract

For a perfect field $k$ of characteristic $p>0$ and for a finite dimensional symmetric $k$-algebra $A$ K\"ulshammer studied a sequence of ideals of the centre of $A$ using the $p$-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If $A$ is symmetric then the dual to the K\"ulshammer ideal structure was generalised to higher Hochschild homology in earlier work. In the present paper we follow this program and propose an analogue of the dual to the K\"ulshammer ideal structure on the degree $m$ Hochschild homology theory also to not necessarily symmetric algebras.