Rickard's Derived Morita Theory: Review and Outlook

Abstract

We survey the main results in Jeremy Rickard's seminal papers `Morita theory for derived categories' and `Derived equivalences and derived functors'. These papers catalysed the later development of the Morita theory of (enhanced) compactly generated triangulated categories by Keller in the algebraic setting and by Schwede and Shipley in the topological setting. We also discuss the role of Rickard's notion of splendid equivalence in the context of Brou\'e's abelian defect group conjecture, and indicate an alternative proof of parts of Rickard's Derived Morita Theorem that leverages the notion of completion of a triangulated category.