Cohen Macaulay Local Ring Research Articles

The bi-canonical degree of a Cohen–Macaulay ring

This paper is a sequel to [8] where we introduced an invariant, called canonical degree, of Cohen–Macaulay local rings that admit a canonical ideal. Here to each such ring R with a canonical ideal, we attach a different invariant, called bi-canonical degree, which in dimension 1 appears also in [12] as the residue of R. The minimal values of these functions characterize specific classes of Cohen–Macaulay rings. We give a uniform presentation of such degrees and discuss some computational opportunities offered by the bi-canonical degree.

CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

Totally reflexive modules and Poincaré series

We study Cohen–Macaulay non-Gorenstein local rings (R,m,k) admitting certain totally reflexive modules. More precisely, we give a description of the Poincaré series of k by using the Poincaré series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen–Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.

On the monotonicity of Hilbert functions

In this paper we show that a large class of one-dimensional Cohen–Macaulay local rings (\mathcal A,\mathfrak m) has the property that if M is a maximal Cohen–Macaulay A -module then the Hilbert function of M (with respect to \mathfrak m ) is non-decreasing. Examples include (1) complete intersections A = Q/(f,g) where (Q,\mathfrak n) is regular local of dimension three and f \in \mathfrak n^2 \setminus \mathfrak n^3 ; (2) one dimensional Cohen–Macaulay quotients of a two dimensional Cohen–Macaulay local ring with pseudo-rational singularity.

Ulrich modules over Cohen–Macaulay local rings with minimal multiplicity

Abstract Let R be a Cohen–Macaulay local ring. In this paper, we study the structure of Ulrich R-modules mainly in the case where R has minimal multiplicity. We explore generation of Ulrich R-modules and clarify when the Ulrich R-modules are precisely the syzygies of maximal Cohen–Macaulay R-modules. We also investigate the structure of Ulrich R-modules as an exact category.

VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Entropy in the category of perfect complexes with cohomology of finite length

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic.Furthermore, given a flat morphism of Cohen–Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is “additive”. Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.

On the vanishing of self extensions over Cohen-Macaulay local rings

The celebrated Auslander-Reiten Conjecture, on the vanishing of self extensions of a module, is one of the long-standing conjectures in ring theory. Although it is still open, there are several results in the literature that establish the conjecture over Gorenstein rings under certain conditions. The purpose of this article is to obtain extensions of such results over Cohen-Macaulay local rings that admit canonical modules. In particular, our main result recovers theorems of Araya, and Ono and Yoshino simultaneously.

The Structure of Chains of Ulrich Ideals in Cohen-Macaulay Local Rings of Dimension One

This paper studies Ulrich ideals in one-dimensional Cohen-Macaulay local rings. A correspondence between Ulrich ideals and overrings is given. Using the correspondence, chains of Ulrich ideals are closely explored. The specific cases where the rings are of minimal multiplicity and \operatorname{GGL} rings are analyzed.

On the computation of the Ratliff-Rush closure, associated graded ring and invariance of a length

Let $(R,\frak{m} )$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\frak{m} $-primary ideal of $R$, and let $J$ be a minimal reduction of $I$. In this paper, we show that, if $\widetilde {I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde {I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that, if $r_J(I)=2$, then $\widetilde {I}=I$ if and only if $\widetilde {I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results of \cite {Cpv, G}. Finally, we give a counter example for Puthenpurakal [Question 3]{P1}.

A filtration of the Sally module and the first normal Hilbert coefficient

The Sally module of an ideal is an important tool to interplay between Hilbert coefficients and the properties of the associated graded ring. In this paper we give new insights on the structure of the Sally module. We apply these results to characterize the almost minimal value of the first Hilbert coefficient in the case of the normal filtration in an analytically unramified Cohen–Macaulay local ring.

Applications and homological properties of local rings with decomposable maximal ideals

We construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R satisfies the uniform Auslander condition (UAC), but the localization Rp does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal p∈Spec(R) such that R has exactly two non-isomorphic semidualizing modules, but the localization Rp has 2n non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type.

DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.

On the Vasconcelos inequality for the fiber multiplicity of modules

ABSTRACTLet (R,𝔪) be a Noetherian local ring of dimension d>0 with infinite residue field. Let M be a finitely generated proper R-submodule of a free R-module F with ℓ(F∕M)<∞ and having rank r. In this article, we study the fiber multiplicity f0(M) of the module M. We prove that if (R,𝔪) is a two-dimensional Cohen–Macaulay local ring, then , where bri(M) denotes the ith Buchsbaum-Rim coefficient of M.

Almost Gorenstein Rees Algebras of pg-Ideals, Good Ideals, and Powers of the Maximal Ideals

Let (A,m) be a Cohen–Macaulay local ring, and let I be an ideal of A. We prove that the Rees algebra R(I) is an almost Gorenstein ring in the following cases: (1) (A,m) is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field K≅A/m, and I is a pg-ideal; (2) (A,m) is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and I=mℓ for all ℓ≥1; (3) (A,m) is a regular local ring of dimension d≥2, and I=md−1. Conversely, if R(mℓ) is an almost Gorenstein graded ring for some ℓ≥2 and d≥3, then ℓ=d−1.

Tensoring with the Frobenius endomorphism

Let $R$ be a commutative Noetherian Cohen-Macaulay local ring that has positive dimension and prime characteristic. Li proved that the tensor product of a finitely generated non-free $R$-module $M$ with the Frobenius endomorphism ${}^{\varphi^n}\!R$ is not maximal Cohen-Macaulay provided that $M$ has rank and $n\gg 0$. We replace the rank hypothesis with the weaker assumption that $M$ is locally free on the minimal prime ideals of $R$. As a consequence, we obtain, if $R$ is a one-dimensional non-regular complete reduced local ring that has a perfect residue field and prime characteristic, then ${}^{\varphi^n}\!R \otimes_{R}{}^{\varphi^n}\!R$ has torsion for all $n\gg0$. This property of the Frobenius endomorphism came as a surprise to us since, over such rings $R$, there exist non-free modules $M$ such that $M\otimes_{R}M$ is torsion-free.

Presentations of rings with a chain of semidualizing modules

Inspired by Jorgensen et al., it is proved that if a Cohen-Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$-modules of length $n$, then $R\cong Q/(I_1+\cdots +I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\dots , I_n$ of $Q$; and, for each $\Lambda \subseteq [n]$, the ring $Q/(\sum _{\ell \in \Lambda } I_\ell )$ has some interesting cohomological properties. This extends the result of Jorgensen et al., and also of Foxby and Reiten.

When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

Let A be a Cohen–Macaulay local ring with dimA=d≥3, possessing the canonical module KA. Let a1,a2,…,ar (3≤r≤d) be a subsystem of parameters of A, and set Q=(a1,a2,…,ar). We show that if the Rees algebra R(Q) of Q is an almost Gorenstein graded ring, then A is a regular local ring and a1,a2,…,ar is a part of a regular system of parameters of A.

Property of Almost Cohen–Macaulay over Extension Modules

Let (R, m) be a Cohen–Macaulay local ring of dimension d, C a canonical R-module and M an almost Cohen–Macaulay R-module of dimension n and of depth t. We prove that dim [Formula: see text] and if [Formula: see text] then [Formula: see text] is an almost Cohen–Macaulay R-module. In particular, if [Formula: see text] then HomR(M, C) is an almost Cohen–Macaulay R-module. In addition, with some conditions, we show that [Formula: see text] is also almost Cohen–Macaulay. Finally, we study the vanishing [Formula: see text] and [Formula: see text].