Related papers: Higher Auslander--Solberg correspondence for exact…

We investigate the structure of certain almost split sequences in $\mathcal{P}(\Lambda)$, i.e., the category of morphisms between projective modules over an Artin algebra $\Lambda$. The category $\mathcal{P}(\Lambda)$ has very nice…

Let $\mathcal{E}=(\mathcal{A},\mathcal{S})$ be an exact category with enough projectives $\mathcal{P}$. We introduce the notion of support $\tau$-tilting subcategories of $\mathcal{E}$. It is compatible with existing definitions of support…

We define the symmetric Auslander category A^s(R) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right tails of totally acyclic complexes of projective modules. The symmetric Auslander…

The aim of the paper is to discuss the relation subgroups of the Grothendieck groups of extriangulated categories and certain other subgroups. It is shown that a locally finite extriangulated category $\C$ has Auslander-Reiten…

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)…

Recently, Wang, Wei and Zhang introduced the notion of recollements of extriangulated categories. In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We provide some methods to construct…

We prove that if the Auslander-Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull-Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of…

Given the pair of a dualizing $k$-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander's…

Let $\mathscr{C}$ be an additive subcategory of left $\Lambda$-modules, we establish relations of the orthogonal classes of $\mathscr{C}$ and (co)res $\widetilde{\mathscr{C}}$ under separable equivalences. As applications, we obtain that…

We establish a $d$-dimensional Auslander correspondence for $d$-truncated proper connective DG-algebras via $d$-extended module categories. A $d$-truncated proper connective DG-algebra $\Gamma$ is called Auslander if its $d$-extended module…

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…

A ring with an Auslander dualizing complex is a generalization of an Auslander-Gorenstein ring. We show that many results which hold for Auslander-Gorenstein rings also hold in the more general setting. On the other hand we give criteria…

Let $\C$ be an $(n+2)$-angulated category with shift functor $\Sigma$ and $\X$ be a cluster-tilting subcategory of $\C$. Then we show that the quotient category $\C/\X$ is an $n$-abelian category. If $\C$ has a Serre functor, then $\C/\X$…

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the…

We show that the relative Auslander-Buchweitz context on a triangulated category $\T$ coincides with the notion of co-$t$-structure on certain triangulated subcategory of $\T$ (see Theorem \ref{M2}). In the Krull-Schmidt case, we stablish a…

For a path algebra $A$ over a quiver $Q$, there are bijections between the support-tilting modules of $A$, torsion classes in $\mathrm{mod}(A)$ and wide subcategories in $\mathrm{mod}(A)$; these are part of the Ingalls-Thomas bijections. As…

We use the theory of Auslander Buchweitz approximations to classify certain resolving subcategories containing a semidualizing or a dualizing module. In particular, we show that if the ring has a dualizing module, then the resolving…

A subcategory $\mathscr{W}$ of an abelian category is called wide if it is closed under kernels, cokernels, and extensions. Wide subcategories are of interest in representation theory because of their links to other homological and…

This paper introduces Auslander-Reiten triangles in subcategories of triangulated categories. The main theorem of the paper shows that there is a close connection with covers and envelopes, also known as minimal right- and…

For a triangulated category T, if C is a cluster-tilting subcategory of T, then the quotient category T\C is an abelian category. Under certain conditions, the converse also holds. This is an very important result of cluster-tilting theory,…