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An Auslander-type result for Gorenstein-projective modules
An artin algebra is said to be CM-finite if there are only finitely many,
up to isomorphisms, indecomposable finitely generated Gorenstein-projective
-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
Generalized Serre duality
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 is Gorenstein if and only if the bounded
homotopy category of finitely generated projective -modules has Serre
duality in the sense of Bondal and Kapranov
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