Generalized Serre duality

Abstract

We introduce a notion of generalized Serre duality on a Hom-finite Krull–Schmidt triangulated category T . This duality induces the generalized Serre functor on T , which is a linear triangle equivalence between two thick triangulated subcategories of T . Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of T containing all the indecomposable objects which appear as the third term of an Auslander–Reiten triangle in T ; dually, the range of the generalized Serre functor is the smallest additive subcategory of T containing all the indecomposable objects which appear as the first term of an Auslander–Reiten triangle in T . 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.