Derived equivalences and stable equivalences of Morita type, II

Abstract

We consider the question of lifting stable equivalences of Morita type to derived equivalences. One motivation comes from an approach to Broué’s abelian defect group conjecture. Another motivation is a conjecture by Auslander and Reiten on stable equivalences preserving the number of non-projective simple modules. A machinery is presented to construct lifts for a large class of algebras, including Frobenius-finite algebras introduced in this paper. In particular, every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. Consequently, the Auslander–Reiten conjecture is true for stable equivalences of Morita type between Frobenius-finite algebras. Examples of Frobenius-finite algebras are abundant, including representation-finite algebras, Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we show that, for a Nakayama-stable idempotent element $e$ in an algebra $A$ over an algebraically closed field, each tilting complex over $eAe$ can be extended to a tilting complex over A that induces an almost ν-stable derived equivalence studied in the first paper of this series. Moreover, the machinery is applicable to verify Broué’s abelian defect group conjecture for several cases mentioned in the literature.