Standard derived equivalences and $~\bf{D}$-standard abelian categories

Abstract

In 1991, Rickard raised the following conjecture: Any derived equivalence between modulecategories is standard, i.e., isomorphic to the derived tensor functor by a two-sided tilting complex.We introduce the notion of a $~\bf{D}$-standard Abelian category, and conjecture that any module categoryis $~\bf{D}$-standard. We survey recent progress on these conjectures.