Derived equivalences and stable equivalences of Morita type, I

TL;DR

This paper explores when derived equivalences between algebras imply stable equivalences of Morita type, extending Rickard's classic result and providing methods to construct such equivalences, revealing shared invariants among related algebras.

Contribution

It establishes conditions under which derived equivalences induce stable equivalences of Morita type and introduces inductive methods for constructing these equivalences.

Findings

Derived equivalences can induce stable equivalences of Morita type under certain conditions.

A functor between stable categories can compare homological dimensions of algebras.

Many non-self-injective algebras are both derived and stably equivalent, sharing invariants.

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 paper, we first show that each derived equivalence $F$ between the derived categories of Artin algebras $A$ and $B$ arises naturally a functor $\bar{F}$ between their stable module categories, which can be used to compare certain homological dimensions of $A$ with that of $B$; and then we give a sufficient condition for the functor $\bar{F}$ 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 classic 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 which are both derived-equivalent and stably equivalent of Morita type, thus they share many common invariants.