Cohen–Macaulay monomial ideals of codimension 2

We give a structure theorem for Cohen–Macaulay monomial ideals of codimension 2, and describe all possible relation matrices of such ideals. In case that the ideal has a linear resolution, the relation matrices can be identified with the spanning trees of a connected chordal graph with the property that each distinct pair of maximal cliques of the graph has at most one vertex in common.     

Graded Cohen–Macaulay rings of wild Cohen–Macaulay type

We give sufficient 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 eventual index of reducibility of parameter ideals and the sequentially Cohen–Macaulay property

Archiv der Mathematik - In this paper, our purpose is to give a characterization of a sequentially Cohen–Macaulay module, which was introduced by Stanley (Combinatorics and Commutative...     

Cohen–Macaulay Loci of Modules

The Cohen–Macaulay locus of any finite module over a noetherian local ring A is studied, and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen–Macaulay are studied.     

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

Syzygies of Cohen–Macaulay Modules over One Dimensional Cohen–Macaulay Local Rings

Algebras and Representation Theory - We study syzygies of (maximal) Cohen–Macaulay modules over one dimensional Cohen–Macaulay local rings. We assume that rings are generically...     

Direct limits of Cohen–Macaulay rings

Let A be a direct limit of a direct system of Cohen–Macaulay rings. In this paper, we describe the Cohen–Macaulay property of A. Our results indicate that A is not necessarily Cohen–Macaulay. We show A is Cohen–Macaulay under various assumptions. As an application, we study Cohen–Macaulayness of non-affine normal semigroup rings.     

When almost Cohen–Macaulay algebras map into big Cohen–Macaulay modules

In this article, we show that almost Cohen–Macaulay algebras are solid. Moreover, we seek for the conditions when (a) an almost Cohen–Macaulay algebra is a phantom extension and (b) when it maps into a balanced big Cohen–Macaulay module.     

Cohen–Macaulay Auslander algebras of skewed-gentle algebras

For any skewed-gentle algebra, we characterize its indecomposable Gorenstein projective modules explicitly and describe its Cohen–Macaulay Auslander algebra. We prove that skewed-gentle algebras are always Gorenstein, which is independent of the characteristic of the ground field, and the Cohen–Macaulay Auslander algebras of skewed-gentle algebras are also skewed-gentle algebras.     

Vanishing of cohomology over Cohen–Macaulay rings

A 2003 counterexample to a conjecture of Auslander brought attention to a family of rings—colloquially called AC rings—that satisfy a natural condition on vanishing of cohomology. Several results attest to the remarkable homological properties of AC rings, but their definition is barely operational, and it remains unknown if they form a class that is closed under typical constructions in ring theory. In this paper, we study transfer of the AC property along local homomorphisms of Cohen–Macaulay rings. In particular, we show that the AC property is preserved by standard procedures in local algebra. Our results also yield new examples of Cohen–Macaulay AC rings.     

Cohen–Macaulay Modules of Infinite Projective Dimension

A characterization of finitely generated torsion modules of not necessarily finite projective dimension over a Cohen–Macaulay ring, is given in terms of the non-Cohen–Macaulay loci and the Fitting invariants of a free resolution of such a module.     

Buchsbaum Stanley–Reisner Rings and Cohen–Macaulay Covers

First, we give a new criterion for Buchsbaum Stanley–Reisner rings to have linear resolutions. Next, we prove that every (d ? 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen–Macaulay complex whose (d ? 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen–Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen–Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h-vectors of the 3-dimensional Buchsbaum Stanley–Reisner rings with initial degree 3.     

Local Rings of Bounded Cohen–Macaulay Type

Let be a complete 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 classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer–Thrall theorem for excellent rings. Presented by J. HerzogMathematics Subject Classifications (2000) 13C05, 13C14, 13H10.     

The classification of special Cohen–Macaulay modules

In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.     

Lyubeznik Table of Sequentially Cohen–Macaulay Rings

We prove that sequentially Cohen–Macaulay rings in positive characteristic, as well as sequentially Cohen–Macaulay Stanley–Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.     

Tensor Products of Approximately Cohen–Macaulay Rings

Our aim in this article is to study a problem originally raised by Grothendieck. We show that the approximately Cohen–Macaulay property is preserved for the tensor product of algebras over a field k. We also discuss the converse problem.     

Some Cohen–Macaulay and Unmixed Binomial Edge Ideals

We study unmixed and Cohen-Macaulay properties of the binomial edge ideal of some classes of graphs. We compute the depth of the binomial edge ideal of a generalized block graph. We also characterize all generalized block graphs whose binomial edge ideals are Cohen–Macaulay and unmixed. So that we generalize the results of Ene, Herzog, and Hibi on block graphs. Moreover, we study unmixedness and Cohen–Macaulayness of the binomial edge ideal of some graph products such as the join and corona of two graphs with respect to the original graphs.