Fine Hochschild invariants of derived categories for symmetric algebras

Abstract

Let $A$ be a symmetric $k$-algebra over a perfect field $k$. Külshammer defined for any integer $n$ a mapping $\zeta \_n$ on the degree 0 Hochschild cohomology and a mapping $\kappa \_n$ on the degree 0 Hochschild homology of $A$ as adjoint mappings of the respective $p$-power mappings with respect to the symmetrizing bilinear form. In an earlier paper it is shown that $\zeta \_n$ is invariant under derived equivalences. In the present paper we generalize the definition of $\kappa \_n$ to higher Hochschild homology and show the invariance of $\kappa$ and its generalization under derived equivalences. This provides fine invariants of derived categories.