Higher Hochschild homology is not a stable invariant
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The purpose of this thesis is to study the higher Hochschild homology of some rational algebras. We know that for some algebras the higher Hochschild homology is a stable invariant. Studying the homology over the spheres and over the torus, we can deduce that this is not true in the most general sense. We will give a counterexample based on the algebra of the dual numbers over a field of characteristic 0. Moreover we will study the equivariant structure of the iterated Hochschild homology for some particular algebras as a toy model, in order to shed some lights on the limits of the topological version of the Hochschild homology, which plays a key role in the understanding of the chromatic shift of K- theory.