Gorenstein derived equivalences and their invariants

Abstract

The main objective of this paper is to study the relative derived categories from various points of view. Let A be an abelian category and C be a contravariantly finite subcategory of A. One can define C-relative derived category of A, denoted by DC*(A). The interesting case for us is when A has enough projective objects and C=GP-A is the class of Gorenstein projective objects, where DC*(A) is known as the Gorenstein derived category of A. We explicitly study the relative derived categories, specially over artin algebras, present a relative version of Rickard's theorem, specially for Gorenstein derived categories, provide some invariants under Gorenstein derived equivalences and finally study the relationships between relative and (absolute) derived categories. © 2013 Elsevier B.V.