Derived equivalences and stable equivalences of Morita type, II

Wei Hu, Changchang Xi

Published 2014-12-23Version 1

Motivated by understanding the Brou\'e's abelian defect group conjecture from algebraic point of view, we consider the question of how to lift a stable equivalence of Morita type between arbitrary finite dimensional algebras to a derived equivalence. In this paper, we present a machinery to solve this question for a class of stable equivalences of Morita type. In particular, we show that every stable equivalence of Morita type between Frobenius-finite algebras over an algebraically closed field can be lifted to a derived equivalence. %Thus Frobenius-finite algebras share many common %invariants of both derived equivalences and stable equivalences. Especially, Auslander-Reiten conjecrure is true for stable equivalences of Morita type between Frobenius-finite algebras without semisimple direct summands. Examples of such a class of algebras are abundant, including Auslander algebras, cluster-tilted algebras and certain Frobenius extensions. As a byproduct of our methods, we further show that, for a Nakayama-stable idempotent element $e$ in an algebra $A$ over an arbitrary field, each tilting complex over $eAe$ can be extended to a tilting complex over $A$ that induces an almost $\nu$-stable derived equivalence studied in the first paper of this series. Moreover, we demonstrate that our techniques are applicable to verify the Brou\'{e}'s abelian defect group conjecture for several cases mentioned by Okuyama.