Cohen-Macaulay Ring Research Articles

Minimal Flat Resolutions of Artinian Modules

The purpose of this paper is to establish connection between the minimal flat resolutions of Artinian modules and the concept of cosequences in commutative algebra. The main result of this paper leads to a characterization of local Cohen–Macaulay rings in terms of vanishing property of the dual Bass numbers of certain Artinian modules.

LENGTH ONE IDEAL EXTENSIONS AND THEIR ASSOCIATED GRADED RINGS

Let (R. m) be a d-dimensional Cohen-Macaulay local ring. Given m-primary ideals J ⊂ I of R such that I is contained in the integral closure of J and λ(I/J)= I, we compare depth G(J) and depth G(J). For example, if J has reduction number one, JI = I2, and μ(J)≤ d + 1, we prove that depth G(I)≥d – 1. If, in addition, μ(I)= d + 1, we show that I has reduction number one, and hence G(I) is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions.

Special homological dimensions and intersection theorem

Let $(R,m)$ be commutative Noetherian local ring. It is shown that $R$ is Cohen-Macaulay ring if there exists a Cohen-Macaulay finite (i.e. finitely generated) $R$-module with finite upper Gorenstein dimension. In addition, we show that, in the Intersection Theorem, projective dimension can be replaced by quasi-projective dimension.

The Hilbert function of the Ratliff–Rush filtration

The Ratliff–Rush filtration has been shown to be a very useful tool for studying numerical invariants of the associated graded ring G ≔ ⊕ t ⩾ 0 ( I t / I t + 1 ) of a local ring ( A , m ) with respect to the classical I -adic filtration. The advantage of this approach is that the associated graded ring G ˜ of A with respect to the Ratliff–Rush filtration has positive depth, but unfortunately G ˜ is not necessarily a standard graded algebra. In this paper, we study some numerical invariants of G ˜ when I is an m -primary ideal of a local Cohen–Macaulay ring and, as consequence, we prove an upper bound on the first coefficient of the Hilbert polynomial of G which extends the already known bounds.

Depth of associated graded rings via Hilbert coefficients of ideals

Given a local Cohen–Macaulay ring ( R , m ) , we study the interplay between the integral closedness—or even the normality—of an m -primary R -ideal I and conditions on the Hilbert coefficients of I . We relate these properties to the depth of the associated graded ring of I .

GORENSTEIN AND Ω-GORENSTEIN INJECTIVE COVERS AND FLAT PREENVELOPES#

ABSTRACT The aim of this paper is to show that if R is a commutative local Cohen–Macaulay ring admitting a dualizing module Ω , then Gorenstein and Ω -Gorenstein injective covers and Gorenstein and Ω -Gorenstein flat preenvelopes exist for certain classes of R -modules.

Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

This paper addresses problems related to the existence of arithmetic Macaulayfications of projective schemes. Let Y be the blow-up of a projective scheme X = ProjR along the ideal sheaf of I ⊂ R. It is known that there are embeddings Y ∼ Projk((I e )c) for c ≥ d(I)e + 1, where d(I) denotes the maximal generating degree of I, and that there exists a Cohen-Macaulay ring of the form k((I e )c) if and only if H 0 (Y, OY ) = k, H i (Y, OY ) = 0 for i = 1,...,dimY − 1, Y is equidimensional and Cohen-Macaulay. Cutkosky and Herzog asked when there is a linear bound on c and e ensuring that k((I e )c) is a Cohen-Macaulay ring. We obtain a surprising compelte answer to this question, namely, that under the above conditions, there are well determined invariants and e0 such that k((I e )c) is Cohen-Macaulay for all c > d(I)e + and e > e0. Our approach is based on recent results on the asymptotic linearity of the Castelnuovo-Mumford regularity of ideal powers. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form R((I e )ct) (which provides an arithmetic Macaulayfication for X). If R has negative a � -invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if ��OY = OX, R i ��OY = 0 for i > 0, Y is equidimensional and Cohen-Macaulay. Especially, these conditions imply the Cohen-Macaulayness of R((I e )ct) for all c > d(I)e + and e > e0. The above results can be applied to obtain several new classes of Cohen-Macaulay algebras.

Ω-Gorenstein Projective and Flat Covers and Ω-Gorenstein Injective Envelopes#

In this paper, we show that if R is a local Cohen–Macaulay ring admitting a dualizing module Ω, then Ω-Gorenstein projective and flat covers and Ω-Gorenstein injective envelopes exist for certain modules. These results generalize the well known results for local Gorenstein rings.

On the depth of the associated graded ring of a filtration

Let F be a filtration of a Cohen–Macaulay ring such that we can define its associated graded ring G ( F ) . We will give a suitable condition on F which enables us to estimate the depth of G ( F ) . It generalizes several known results on associated graded rings of ideals.

Foxby Equivalence and Cotorsion Theories relative to Semi-dualizing Modules

Foxby duality has proven to be an important tool in studying the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. Recently the notion of a semi-dualizing module has been given [2]. Given a semi-dualizing module the relative Foxby classes can be defined and there is still an associated Foxby duality. We consider these classes (separately called the Auslander and Bass classes) and two naturally defined subclasses which are equivalent to the full subcategories of injective and flat modules. We consider the question of when these subclasses form part of one of the two classes of a cotorsion theory. We show that when this is the case, the associated cotorsion theory is not only complete but in fact is perfect. We show by examples that even when the semi-dualizing module is in fact dualizing over a local Cohen-Macaulay ring it both may or may not occur that we get this associated cotorsion theory.

The depth of the associated graded ring of ideals with any reduction number

Let R be a local Cohen–Macaulay ring, let I be an R-ideal, and let G be the associated graded ring of I. We give an estimate for the depth of G when G is not necessarily Cohen–Macaulay. We assume that I is either equimultiple, or has analytic deviation one, but we do not have any restriction on the reduction number. We also give a general estimate for the depth of G involving the first r+ℓ powers of I, where r denotes the Castelnuovo regularity of G and ℓ denotes the analytic spread of I.

On birational Macaulayfications and Cohen–Macaulay canonical modules

The aim of the paper is twofold. At first there is a characterization of those local domains admitting a birational Macaulayfication. It turns out that a local domain A, quotient of a Gorenstein ring, possesses a Cohen–Macaulay birational extension B if and only if its canonical module K( A) is a Cohen–Macaulay module. In this situation it follows that B is isomorphic to the endomorphism ring of K( A). The second part of the paper is devoted to the question when the canonical module of a local ring is a Cohen–Macaulay module. There are applications on sequentially Cohen–Macaulay rings. A large class of rings whose canonical module are Cohen–Macaulay is given by the simplicial affine semigroup rings. As a technical tool there is a spectral sequence related to the duality with respect to the dualizing complex.

Gorenstein homological dimensions

In basic homological algebra, the projective, injective and flat dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein flat dimensions are studied. There is a variety of nice results about Gorenstein dimensions over special commutative noetherian rings; very often local Cohen–Macaulay rings with a dualizing module. These results are done by Avramov, Christensen, Enochs, Foxby, Jenda, Martsinkovsky and Xu among others. The aim of this paper is to generalize these results, and to give homological descriptions of the Gorenstein dimensions over arbitrary associative rings.

Core versus graded core, and global sections of line bundles

We find formulas for the graded core of certainm\mathfrak {m}-primary ideals in a graded ring. In particular, ifSSis the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal ofSSgenerated by all elements of degrees at leastNN(for some, equivalently every, largeNN) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.

Graded local cohomology modules and their associated primes: the Cohen–Macaulay case

It is well-known that the ith local cohomology of a finitely generated R-module M over a positively graded commutative Noetherian ring R, associated to the irrelevant ideal R + is graded. Furthermore, for every integer n, the nth component H R + i ( M) n of this local cohomology module H R + i ( M) is finitely generated over R 0 and vanishes for n⪢0. In this paper, we want to understand the behavior of H R + i ( M) n for n⪡0 in the case where R is a Cohen–Macaulay ring and M is a Cohen–Macaulay R-module. When dim R 0=1 , we will show that Ass R 0 ( H R + i ( M) n ) becomes constant when n becomes negatively large. When R 0 is local, and dim R 0=2 , we will show that there exists an integer N such that either H R + i ( M) n =(0) for all n< N or, H R + i ( M) n ≠(0) for all n< N.

Generalized Cohen–Macaulay dimension

A new homological dimension, called GCM-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. GCM-dimension (short for Generalized Cohen–Macaulay dimension) characterizes Generalized Cohen–Macaulay rings in the sense that: a ring R is Generalized Cohen–Macaulay if and only if every finitely generated R-module has finite GCM-dimension. This dimension is finer than CM-dimension and we have equality if CM-dimension is finite. Our results will show that this dimension has expected basic properties parallel to those of the homological dimensions. In particular, it satisfies an analog of the Auslander–Buchsbaum formula. Similar methods will be used for introducing quasi-Buchsbaum and Almost Cohen–Macaulay dimensions, which reflect corresponding properties of their underlying rings.

Hilbert coefficients and Buchsbaumness of associated graded rings

Let A be a Noetherian local ring with the maximal ideal m and I an m -primary ideal. The purpose of this paper is to generalize Northcott's inequality on Hilbert coefficients of I given in Northcott (J. London Math. Soc. 35 (1960) 209), without assuming that A is a Cohen–Macaulay ring. We will investigate when our inequality turns into an equality. It is related to the Buchsbaumness of the associated graded ring of I.

On a Conjecture of R. P. Stanley; Part II--Quotients Modulo Monomial Ideals

In 1982 Richard P. Stanley conjectured that any finitely generated \Bbb Zn-graded module M over a finitely generated \Bbb Nn-graded \Bbb K-algebra R can be decomposed as a direct sum M e ⊕i e 1t νiSi of finitely many free modules νiSi which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the Si have to be subalgebras of R of dimension at least depth M. We will study this conjecture for modules M e R/I, where R is a polynomial ring and I a monomial ideal. In particular, we will prove that Stanley's Conjecture holds for the quotient modulo any generic monomial ideal, the quotient modulo any monomial ideal in at most three variables, and for any cogeneric Cohen-Macaulay ring. Finally, we will give an outlook to Stanley decompositions of arbitrary graded polynomial modules. In particular, we obtain a more general result in the 3-variate case.

Some results on associated primes of local cohomology modules

The paper contributes to the question whether the set of associated prime ideals of the local cohomology module HiI (M) is finite for all ideals I of a local ring (R, m) and a finitely generated generalized Cohen-Macaulay R-module M. We prove that it will be enough to solve the problem for i=2 resp. 3 for ideals with two resp. certain ideals with three generators. This extends Hellus' result, see [5], of a Cohen-Macaulay ring. Moreover, in the case of M a Cohen-Macaulay module there is another sufficient criterion for the finiteness of associated prime ideals of HiI (M) related to certain cofiniteness conditions. Finally, we discuss several examples of the literature related to the problem.

STRUCTURE OF THE FLAT COVERS OF ARTINIAN MODULES

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form <TEX>$\prod_{p\epsilonAtt_R(A)}T-p$</TEX>, where <TEX>$Tp$</TEX> is the completion of a free <TEX>R$_{p}$</TEX>-module. Also, we construct a minimal flat resolution for R/xR-module 0: <TEX>$_AX$</TEX> from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = <TEX>$\chiA$</TEX>. This result leads to a description of the structure of a minimal flat resolution for <TEX>${H^n}_{\underline{m}}(R)$</TEX>, nth local cohomology module of R with respect to the ideal <TEX>$\underline{m}$</TEX>, over a local Cohen-Macaulay ring (R, <TEX>$\underline{m}$</TEX>) of dimension n.