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    Generalized Serre duality

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    We introduce a notion of generalized Serre duality on a Hom-finite Krull-Schmidt triangulated category T\mathcal{T}. This duality induces the generalized Serre functor on T\mathcal{T}, which is a linear triangle equivalence between two thick triangulated subcategories of T\mathcal{T}. Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of T\mathcal{T} containing all the indecomposable objects which appear as the third term of an Auslander-Reiten triangle in T\mathcal{T}; dually, the range of the generalized Serre functor is the smallest additive subcategory of T\mathcal{T} containing all the indecomposable objects which appear as the first term of an Auslander-Reiten triangle in T\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 AA is Gorenstein if and only if the bounded homotopy category of finitely generated projective AA-modules has Serre duality in the sense of Bondal and Kapranov

    An Auslander-type result for Gorenstein-projective modules

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    An artin algebra AA is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective AA-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type (\cite{A,A1}).Comment: Comments are welcome. Adv. Math., accepte
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