  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.
