Generalized Serre duality

Abstract: We introduce a notion of generalized Serre duality on a Hom-finite Krull–Schmidt triangulated category . This duality induces the generalized Serre functor on , which is a linear triangle equivalence between two thick triangulated subcategories of . Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of containing all the indecomposable objects which appear as the third term of an Auslander–Reiten triangle in ; dually, the range of the generalized Serre functor is the smallest additive subcategory of containing all the indecomposable objects which appear as the first term of an Auslander–Reiten triangle in . We compute explicitly the generalized Serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of Artin and Zhang. We obtain a characterization of Gorenstein algebras: an artin algebra A is Gorenstein if and only if the bounded homotopy category of finitely generated projective A-modules has Serre duality in the sense of Bondal and Kapranov. [ABSTRACT FROM AUTHOR]