2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/110196We discuss the homological algebra of representation theory of finite dimensional algebras and finite groups. We present various methods for the construction and the study of equivalences of derived categories: local group theory, geometry and categorifications.Article for the Proceedings of ICM 2006 Madrid (31 pages)Representation Theorytext