On iterated almost $\nu$-stable derived equivalences

In a recent paper \cite{HuXi3}, we introduced a classes of derived equivalences called almost $\nu$-stable derived equivalences. The most important property is that an almost $\nu$-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 $\nu$-stable derived equivalences and their quasi-inverses, which is called iterated almost $\nu$-stable derived equivalences. We give a sufficient and necessary condition for a derived equivalence to be an iterated almost $\nu$-stable derived equivalence, and give an explicit construction of the stable equivalence functor induced by an iterated almost $\nu$-stable derived equivalence. As a consequence, we get some new sufficient conditions for a derived finite-dimensional algebras to induce a stable equivalence of Morita type.