J. Gomez-Torrecillas, F. J. Lobillo
In this paper we use refiltering methods to prove that certain types of multi-filtered algebras are Auslander-regular and Cohen-Macaulay. This is applied to obtain that the quantized enveloping algebra associated to a Cartan matrix is Auslander-regular and Cohen-Macaulay.
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The Cohen-Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Furthermore, we show that a class of special string algebras, which are called to be string algebras with G-condition, are representation-finite if and only if their Cohen-Macaulay Auslander algebras are representation-finite. Finally, the self-injective dimension of gentle algebras is characterized using their Cohen-Macaulay Auslander algebras.
A commutative local Cohen-Macaulay ring R of finite Cohen-Macaulay type is known to be an isolated singularity; that is, Spec(R)-m is smooth. This paper proves a non-commutative analogue. Namely, if A is a (non-commutative) graded AS Cohen-Macaulay algebra which is FBN and has finite Cohen-Macaulay type, then the non-commutative projective scheme determined by A is smooth.
We calculate the Krull--Gabriel dimension of the functor category of the (stable) category of maximal Cohen--Macaulay modules over hypersurfaces of countable Cohen--Macaulay representation type. We show that the Krull--Gabriel dimension is $0$ if the hypersurface is of finite Cohen--Macaulay representation type and that is $2$ if the hypersurface is of countable but not finite Cohen--Macaulay representation type.
Auslander-Reiten's Cohen-Macaulay algebras and contracted preprojective algebras
本文通过证明所有Dynkin类型的收缩预投射代数均为Cohen-Macaulay代数，给出了首个非平凡的Cohen-Macaulay代数类，并否定了Auslander-Reiten关于CM模范畴与$d$次合冲范畴是否一致的猜想。
We give suffcient conditions for a standard graded Cohen-Macaulay ring, or equivalently, an arithmetically Cohen-Macaulay projective variety, to be Cohen-Macaulay wild in the sense of representation theory. In particular, these conditions are applied to hypersurfaces and complete intersections.
The purpose of this paper is to present a family of Cohen-Macaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.
The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties
本文证明射影空间中具有 Cohen-Macaulay 分次坐标环的约化闭子概形几乎都是野 Cohen-Macaulay 型，并对少数例外情形给出完全分类。
We study properties of graded maximal Cohen-Macaulay modules over an N-graded locally finite, Auslander Gorenstein, and Cohen-Macaulay algebra of dimension two. As a consequence, we extend a part of McKay correspondence in dimension two to a more general setting.
We study the degeneration problem for maximal Cohen-Macaulay modules and give several examples of such degenerations. It is proved that such degenerations over an even-dimensional simple hypersurface singularity of type $(A_n)$ are given by extensions. We also prove that all extended degenerations of maximal Cohen-Macaulay modules over a Cohen-Macaulay complete local algebra of finite representation type are obtained by iteration of extended degenerations of Auslander-Reiten sequences.
This paper gives a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies two famous approximation theorems; one is due to Auslander and Bridger and the other is due to Auslander and Buchweitz. Modules admitting such approximations shall be studied.
Let $A$ be a Cohen-Macaulay normal domain. A non commutative crepant resolution (NCCR) of $A$ is an $A$-algebra $Γ$ of the form $Γ= End_A(M)$, where $M$ is a reflexive $A$-module, $Γ$ is maximal Cohen-Macaulay as an $A$-module and $gldim(Γ)_P = \dim A_P $ for all primes $P$ of $A$. We give bountiful examples of equi-characteristic Cohen-Macaulay normal local domains and mixed characteristic Cohen-Macaulay normal local domains having NCCR. We also give plentiful examples of affine Cohen-Macaulay normal domains having NCCR.
We give concrete DG-descriptions of certain stable categories of maximal Cohen-Macaulay modules. This makes in possible to describe the latter as generalized cluster categories in certain cases.
Maximal Cohen-Macaulay DG-complexes
本文提出极小Cohen-Macaulay DG复形的定义，扩展了Iyengar等人关于局部环的极小Cohen-Macaulay复形的研究，并建立了具有恒定振幅的DG环的导出改进新交集定理。
On Cohen--Macaulay modules over the Weyl algebra
本文提出了 Weyl 代数上 Cohen--Macaulay 模的定义，并给出了 GKZ $A$-超几何 $D$-模满足 Cohen--Macaulay 性质的充分条件。
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) categories of finite dimensional algebras.
One of the first remarkable results in the representation theory of artin algebras, due to Auslander and Ringel-Tachikawa, is the characterization of when an artin algebra is representation-finite. In this paper, we investigate aspects of representation-finiteness in the general context of exact categories in the sense of Quillen. In this framework, we introduce "big objects" and prove an Auslander-type "splitting-big-objects" theorem. Our approach generalises and unifies the known results from the literature. As a further application of our methods, we extend the theorems of Auslander and Rin…
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