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We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gorenstein injective dimension. [ABSTRACT FROM AUTHOR]

Abstract: We introduce the notion of totally reflexive extension of rings. It unifies Gorenstein orders and Frobenius extensions. We prove that for a totally reflexive extension, a module over the extension ring is totally reflexive if and only if its underlying module over the base ring is totally reflexive. [Copyright &y& Elsevier]

Abstract: 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 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. [ABSTRACT FROM AUTHOR]

Abstract: We prove that triangulated categories with bounded t-structures are Karoubian. Consequently, for an Ext-finite abelian category over a commutative noetherian complete local ring, its bounded derived category is Krull–Schmidt. [Copyright &y& Elsevier]