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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]

Let $R$ be a two-sided Noetherian ring, and let $M$ be a nilpotent $R$ -bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_{R}(M)$. Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_{R}(M)$. We characterize Gorenstein projective $T_{R}(M)$ -modules in terms of $R$ -modules. [ABSTRACT FROM AUTHOR]

We exploit singular equivalences between artin algebras that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra to the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras. It turns out that this singular equivalence restricts to a triangle equivalence between their stable categories of Gorenstein-projective modules, and thus induces a triangle equivalence between their Gorenstein defect categories. [ABSTRACT FROM AUTHOR]

We prove that a certain pair of bimodules over two artin algebras gives rise to a triangle equivalence between the singularity categories of the two corresponding trivial extension algebras. Some consequences and an example are given. [ABSTRACT FROM PUBLISHER]

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: The aim of this paper is to construct comodules of and modules of via quiver, where is not a root of unity. By embedding into the path coalgebra , where is the Gabriel quiver of as a coalgebra, we obtain a basis of in terms of combinations of paths of the quiver ; this special basis enables us to describe the category of -comodules by certain representations of ; and this description further permits us to construct a class of modules of , from certain representations of , via the duality between and . [Copyright &y& Elsevier]

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]