Related papers: Cleft extensions of rings and singularity categori…

We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let ${\omega}$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion…

We study totally acyclic complexes of projective modules over triangular matrix rings and then use it to classify Gorenstein projective modules over such rings. We also use this classification to obtain some information concerning…

Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…

Given an abelian category, we introduce a categorical concept of (strongly) Gorenstein projective (resp., injective) objects, by defining a new special class of objects. Then we study the transfer of these properties when passing to an…

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology…

A ring $R$ is called left GF-closed, if the class of all Gorenstein flat left $R$-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak…

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the…

We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective…

This is a survey on recent developments in Cohen-Macaulay representations via tilting and cluster tilting theory. We explain triangle equivalences between the singularity categories of Gorenstein rings and the derived (or cluster)…

In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable…

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two…

We establish a Morita theorem to construct triangle equivalences between the singularity categories of (commutative and non-commutative) Gorenstein rings and the cluster categories of finite dimensional algebras over fields, and more…

In this paper, we compare the Gorenstein homological dimension of a ring $R$ and of its trivial ring extension by an module $E$.

New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…

We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with…

For a tensor ring $T_R(M)$, under certain conditions, we characterize the Gorenstein projective modules over $T_R(M)$, and prove that a $T_R(M)$-module $(X,u)$ is Gorenstein projective if and only if $u$ is monomorphic and ${\rm coker}(u)$…

This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437--445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of…

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…

In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for…

In this paper, we investigate the behavior of Igusa-Todorov distances, extension and Rouquier dimensions under cleft extensions of abelian categories. We apply our results to Morita context rings, trivial extension rings, tensor rings and…