Derived equivalences and stable equivalences of Morita type, I

Abstract

For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalence $F$ between the derived categories of Artin algebras $A$ and $B$ arises naturally as a functor $\overline{F}$ between their stable module categories, which can be used to compare certain homological dimensions of $A$ with that of $B$. We then give a sufficient condition for the functor $\overline{F}$ to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.