Generalized Serre duality

We introduce the generalized Serre functor S on a skeletally-small Hom-finite Krull-Schmidt triangulated category C. We prove that its domain Cr and range Cl are thick triangulated subcategories. Moreover, the subcategory Cr (resp. Cl) is the smallest additive subcategory containing all the objects in C which appears as the third term (resp. the first term) of some Aulsander-Reiten triangle in C, and the functor S is a triangle equivalence between Cr and Cl. We also compute explicitly the generalized Serre structures on the bounded derived categories of finite-dimensional algebras and certain noncommutative projective schemes. As a byproduct, a seemingly new characterization of Gorenstein algebras is given: a finite-dimensional algebra A is Gorenstein if and only if the bounded homotopy category K(A-proj) of finitely-generated projective A-modules has Serre duality.