On iterated almost $\nu$-stable derived equivalences

TL;DR

This paper studies the composition of almost $ u$-stable derived equivalences, providing conditions for when they are iterated and constructing the associated stable equivalence functors, thus extending known results in derived and stable equivalences.

Contribution

It introduces the concept of iterated almost $ u$-stable derived equivalences, characterizes when a derived equivalence is such, and constructs the corresponding stable functors, expanding the understanding of derived-stable relationships.

Findings

Provides necessary and sufficient conditions for iterated almost $ u$-stable derived equivalences.

Constructs explicit stable equivalence functors from these iterated equivalences.

Offers new criteria for derived finite-dimensional algebras to induce stable equivalences of Morita type.

Abstract

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.