A note on derived categories of the second kind

Abstract

Derived categories of the second kind arise in the study of relative singularities, matrix factorizations for Landau-Ginzburg models, Koszul duality, and infinite tilting theory. Derived categories of the second kind have been defined in terms of absolutely acyclic complexes, a subclass of acyclic complexes depending on totalized double complexes. The paper provides a simple and natural description of derived categories of both kinds in the framework of exact categories, obtaining all derived categories as localizations of exact categories by distinguished biresolving subcategories. Some special triangle equivalences are proved by using this approach.