Related papers: Auslander's Theorem and n-Isolated Singularities

Let R be a commutative noetherian local ring. As an analogue of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated R-modules is introduced in this…

For a finitely generated module $ M $ over a commutative Noetherian ring $R$, we settle the Auslander-Reiten conjecture when at least one of ${\rm Hom}_R(M,R)$ and ${\rm Hom}_R(M,M)$ has finite injective dimension. A number of new…

We study the K'-theory of a CM Henselian local ring R of finite Cohen-Macaulay type. We first describe a long exact sequence involving the groups $K_i'(R)$ and the K-groups of certain other rings, including the Auslander algebra. By…

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the…

Let R be a Cohen-Macaulay ring and M a maximal Cohen-Macaulay R-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global…

Let $\Lambda$ be a finite dimensional Auslander algebra. For a $\Lambda$-module $M$, we prove that the projective dimension of $M$ is at most one if and only if the projective dimension of its socle soc\,$M$ is at most one. As an…

For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = End_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can…

We construct a local Cohen-Macaulay ring $R$ with a prime ideal $\mathfrak{p}\in\spec(R)$ such that $R$ satisfies the uniform Auslander condition (UAC), but the localization $R_{\mathfrak{p}}$ does not satisfy Auslander's condition (AC).…

Let (R,m,k) be a local Cohen-Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will…

We utilize recent results of Andr\'e and Gabber on the existence of weakly functorial integral perfectoid big Cohen-Macaulay (BCM) algebras to study singularities of local rings in mixed characteristic. In particular, we introduce a mixed…

We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and…

Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $\Delta_R(A)$ of $A$ has a number of pleasant properties, such as being…

In this note we first study regular $\mathbb{Z}$-graded local rings. We characterize commutative noetherian regular $\mathbb{Z}$-graded local rings in similar ways as in the usual local case. Then, we characterize graded isolated…

We consider the converse of the Butler, Auslander-Reiten's Theorem which is on the relations for Grothendieck groups. We show that a Gorenstein ring is of finite representation type if the Auslander-Reiten sequences generate the relations…

We study Artin algebras $A$ and commutative Noetherian complete local rings $R$ in connection with the following decomposition property of Gorenstein-projective modules: $(*)$ any Gorenstein-projective module is a direct sum of finitely…

We extend the Auslander-Buchweitz axioms and prove Cohen-Macaulay approximation results for fibred categories. Then we show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of Cohen-Macaulay…

It is well known that for Auslander algebras, the category of all (finitely generated) projective modules is an abelian category and this property of abelianness characterizes Auslander algebras by Tachikawa theorem in 1974. Let $n$ be a…

In this paper, we aim to obtain some results under the condition that the dual of a module over a commutative Noetherian ring has finite Gorenstein dimension. In this direction, we derive results involving vanishing of Ext as well as the…

Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on…

We shall show that the stable categories of graded Cohen-Macaulay modules over quotient singularities have tilting objects. In particular, these categories are triangle equivalent to derived categories of finite dimensional algebras. Our…