Higher Hochschild homology is not a stable invariant.

Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients—which usually is that of abelian groups—is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory. [ABSTRACT FROM AUTHOR]