Cohen Macaulay Local Ring Research Articles

On the first normalized Hilbert coefficient

The main purpose of this paper is to establish upper bounds for the first normalized Hilbert coefficient of an m -primary ideal of a Cohen–Macaulay local ring in terms of its multiplicity and number of generators.

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.

STRONGLY TORSION-FREE MODULES AND LOCAL COHOMOLOGY OVER COHEN–MACAULAY RINGS#

ABSTRACT In this paper we investigate the relationship between strongly torsion-free, maximal Cohen–Macaulay, and strongly cotorsion modules over Cohen–Macaulay local rings via local cohomology. Results are given analogous to those known for Gorenstein flat, totally reflexive, and Gorenstein injective modules over a Gorenstein local ring.

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 .

Some criteria for the Gorenstein property

This paper gives criteria for a Cohen–Macaulay local ring to be Gorenstein, in terms of the vanishing of Ext modules. The main results extend earlier work of Ulrich.

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.

Hilbert coefficients and depth of fiber cones

Criteria are given in terms of certain Hilbert coefficients for the fiber cone F ( I ) of an m -primary ideal I in a Cohen–Macaulay local ring ( R , m ) so that it is Cohen–Macaulay or has depth at least dim ( R ) - 1 . A version of Huneke's fundamental lemma is proved for fiber cones. Goto's results concerning Cohen–Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.

Column Invariants Over Cohen–Macaulay Local Rings

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). #Communicated by I. Swanson.

Fiber Cones of Ideals with Almost Minimal Multiplicity

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi’s bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

Gorenstein Injective Dimension and Dual Auslander–Bridger Formula#

An R-module M satisfies the dual Auslander–Bridger formula if Gid R M + Tor − depth M = depth R. Let R be a complete Cohen–Macaulay local ring with residue field k and M be a non-injective Ext-finite R-module such that for all i ≥ 1 and all indecomposable injective R-modules E ≠ E(k). If M has finite Gorenstein injective dimension, then we will prove that M satisfies the dual Auslander–Bridger formula if either Ext-depth M > 0 or Ext-depth M = 0 and Gid R M ≠ 1. We denote Gorenstein injective envelope of M by G(M). If R is a Gorenstein local ring and M is a non-Gorenstein injective finitely generated R-module and G(M) is reduced, then this formula holds for and its cosyzygy. As the last result, if R is a regular local ring, then dual Auslander–Bridger holds for any non-injective Ext-finite R − module.

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 core of ideals

of I, denoted bycore(I), is defined to be the intersection of all reductions of I.The core of ideals was first studied by Rees and Sally [RS], partly due to itsconnection to the theorem of Brianc¸on and Skoda. Later, Huneke and Swanson[HuS] determined the core of integrally closed ideals in two-dimensional regularlocal rings and showed a close relationship to Lipman’s adjoint ideal. Recently,Corso, Polini and Ulrich [CPU1,2] gave explicit descriptions for the core of certainideals in Cohen-Macaulay local rings, extending the result of [HuS]. In these twopapers, several questions and conjectures were raised which provided motivationfor our work. More recently, Hyry and Smith [HyS] have shown that the core andits properties are closely related to a conjecture of Kawamata on the existence ofsections for numerically effective line bundles which are adjoint to an ample linebundle over a complex smooth algebraic variety, and they generalize the result in[HuS] to arbitrary dimension and more general rings. Nonetheless, there are manyunanswered questions on the nature of the core. One reason is that it is difficult todetermine the core and there are relatively few computed examples.Our focus in this paper is in effective computation of the core with an eye topartially answering some questions raised in [CPU1,2]. A first approach to under-standing the core was given by Rees and Sally. For an ideal Iin a local Noetherianring (R,m)having analytic spread l, one can take lgeneric generators ofIin aringofthe form R[U

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.

Characterizing Cohen-Macaulay local rings by Frobenius maps

Let R be a commutative noetherian local ring of prime characteristic. Denote by e R the ring R regarded as an R-algebra through e-times composition of the Frobenius map. Suppose that R is F-finite, i.e., 1 R is a finitely generated R-module. We prove that R is Cohen-Macaulay if and only if the R-modules e R have finite Cohen-Macaulay dimensions for infinitely many integers e.

Strongly torsion free, copure flat and Matlis reflexive modules

In this paper we show that if R is a ring of Krull dimension d and M is a strongly copure flat (respectively, strongly copure injective) module, then E⊗ R M (respectively, Hom R ( E, M)) is strongly cotorsion (respectively, strongly torsion free) for any injective module E. We obtain a new characterization for copure flat modules. We prove that M is copure flat if and only if Ext R 1( M, F)=0 for all flat cotorsion modules F. Moreover, if ( R, m ) is a Cohen–Macaulay local ring and M is strongly copure flat, then Hom R ( M, F) is strongly torsion free. Let ( R, m ) be a Cohen–Macaulay local ring of Krull dimension d and M be a Matlis reflexive strongly torsion free R-module. We show that M ̂ is a maximal Cohen–Macaulay R ̂ -module. Also, if R is as above and M a Matlis reflexive strongly copure injective R-module, then Hom R(E(R/ m),M) is either zero or a maximal Cohen–Macaulay R ̂ -module.

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 the computation of the Ratliff–Rush closure

Let R be a Cohen–Macaulay local ring with maximal ideal m . In this paper we present a procedure for computing the Ratliff–Rush closure of an m -primary ideal I⊂ R.

Numerical equivalence defined on Chow groups of Noetherian local rings

In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional ${\Bbb Q}$-vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendieck's standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen-Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert-Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters.

Bounds on the a-invariant and reduction numbers of ideals

Let R be a d-dimensional standard graded ring over an Artinian local ring. Let M be the unique maximal homogeneous ideal of R. Let h i ( R) n denote the length of the nth graded component of the local cohomology module H i M (R) . Define the Eisenbud–Goto invariant EG( R) of R to be the number ∑ q=0 d−1 d−1 q h q M (R) 1−q. We prove that the a-invariant a( R) of the top local cohomology module H M d(R) satisfies the inequality: a( R)⩽ e( R)−ℓ( R 1)+( d−1)(ℓ( R 0)−1)+ EG( R). This bound is used to get upper bounds for the reduction number of an m -primary ideal I of a Cohen–Macaulay local ring (R, m) , when the associated graded ring of I has depth at least d−1.

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.