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...     

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.     

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.     

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.     

Column Invariants Over Cohen–Macaulay Local Rings

Abstract

We define some numerical invariants over Cohen–Macaulay local rings. These invariants are related to columns of the presenting matrices of maximal Cohen–Macaulay modules and syzygy modules without free summands. We study the relationship between these invariants, and the invariant col(A).

     

Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

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 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.     

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.     

Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[_x1,…,_xn]$R[x_1,\dots ,x_n]$ generated by all paths of a fixed length ? of T  . We classify all trees for which R/I$R/I$ is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals.     

The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

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.     

(Bi-)Cohen–Macaulay Simplicial Complexes and Their Associated Coherent Sheaves

ABSTRACT

Via the BGG correspondence, a simplicial complex Δ on [n] is transformed into a complex of coherent sheaves on P ^n?1. We show that this complex reduces to a coherent sheaf ? exactly when the Alexander dual Δ* is Cohen–Macaulay.

We then determine when both Δ and Δ* are Cohen–Macaulay. This corresponds to ? being a locally Cohen–Macaulay sheaf.

Lastly, we conjecture for which range of invariants of such Δ's it must be a cone, and show the existence of such Δ's which are not cones outside of this range.     

Cohen–Macaulay,Gorenstein, complete intersection and regular defect for the tensor product of algebras

The main goal of this paper is to measure the defect of Cohen–Macaulay, Gorenstein, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components. Our results allow to generalize various theorems in this topic especially [4, Theorem 2.1], [21, Theorem 6] and [14, Theorems 1 and 2] as well as two Grothendieck's theorems on the transfer of Cohen–Macaulayness and regularity to tensor products over a field issued from finite field extensions. To prove our theorems on the defect of complete intersection and regularity, the homology theory introduced by André and Quillen for commutative rings turns out to be an adequate and efficient tool in this respect.     

Gorenstein Injective Dimension for Complexes and Iwanaga–Gorenstein Rings

The main purpose of this article is to present some applications of the notion of Gorenstein injective dimension of complexes over an associative ring. Among the applications, we give some new characterizations of Iwanaga–Gorenstein rings. In particular, we show that an associative ring R is Iwanaga–Gorenstein if and only if the class of complexes of Gorenstein injective dimension less than or equal to zero and the class of complexes of finite projective dimension are orthogonal complement of each other with respect to the ‘Ext’ functor.     

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...     

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.