Generalized Serre duality

Details Generalized Serre duality Generalized Serre duality

2006-10-08

arxiv.org/abs/math/0610258v3

Journal of Algebra, 328 (2011), 268-286

Mathematics

Representation Theory

Scientific paper

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

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