On iterated almost ν-stable derived equivalences

In a recent paper [5], we introduced a classes of derived equivalences called almost ν-stable derived equivalences. The most important property is that an almost ν-stable derived equivalence always induces a stable equivalence of Morita type, which generalizes a well-known result of Rickard: derived-equivalent self-injective algebras are stably equivalent of Morita type. In this paper, we shall consider the compositions of almost ν-stable derived equivalences and their quasi-inverses, which is called iterated almost ν-stable derived equivalences. We give a sufficient and necessary condition for a derived equivalence to be an iterated almost ν-stable derived equivalence. As a consequence, we get a new sufficient condition for a derived equivalence between general finite-dimensional algebras to induce a stable equivalence of Morita type.