Related papers: Rings with Auslander Dualizing Complexes
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…
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…
Let $(R, \mathfrak{m})$ be a Noetherian local ring. In this paper, we introduce a dual notion for dualizing modules, namely codualizing modules. We study the basic properties of codualizing modules and use them to establish an equivalence…
Let $(R,\fm,k)$ be a commutative noetherian local ring with dualizing complex $\dua R$, normalized by $\Ext^{\depth(R)}_R(k,\dua R)\cong k$. Partly motivated by a long standing conjecture of Tachikawa on (not necessarily commutative)…
We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing…
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…
A version of Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite dimensional Lie…
Let $A$ be an artin algebra. We show that the bounded homotopy category of finitely generated right $A$-modules has Auslander-Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [H2];…
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…
In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these…
In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the…
We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the…
In this article we investigate when a complete ideal in a two-dimensional regular local ring is a multiplier ideal of some ideal with an integral multiplying parameter. In particular, we show that this question is closely connected to the…
An Auslander-Reiten formula for complexes of modules is presented. This formula contains as a special case the classical Auslander Reiten formula. The Auslander-Reiten translate of a complex is described explicitly, and various applications…
We discuss an interesting duality known to occur for certain complex reflection groups, namely the duality groups. Our main construction yields a concrete, representation theoretic realisation of this duality. This allows us to naturally…
A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen-Macaulay ring $R$, the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might…
A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander-Reiten triangles for the $\mathfrak{p}$-local…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…
A "toric face ring", which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied by Bruns, Roemer and their coauthors recently. In this paper, under the "normality" assumption, we describe a dualizing complex of a…
This paper is devoted to study the relationship between two important notions in ring theory, category theory, and representation theory of Artin algebras; namely, Gabriel topologies and higher Auslander(-Gorenstein) algebras. It is…