Cohen Macaulay Local Ring Research Articles

Invariants of Cohen–Macaulay rings associated to their canonical ideals

The purpose of this paper is to introduce new invariants of Cohen–Macaulay local rings. Our focus is the class of Cohen–Macaulay local rings that admit a canonical ideal. Attached to each such ring R with a canonical ideal C, there are integers–the type of R, the reduction number of C–that provide valuable metrics to express the deviation of R from being a Gorenstein ring. We enlarge this list with other integers–the roots of R and several canonical degrees. The latter are multiplicity based functions of the Rees algebra of C.

Faltings’ Finiteness Dimension of Local Cohomology Modules Over Local Cohen–Macaulay Rings

AbstractLet (R, m) denote a local Cohen–Macaulay ring and I a non-nilpotent ideal of R. The purpose of this article is to investigate Faltings’ finiteness dimension fI(R) and the equidimensionalness of certain homomorphic images of R. As a consequence we deduce that fI(R) = max{1, ht I}, and if mAssR(R/I) is contained in AssR(R), then the ring is equidimensional of dimension dim R−1. Moreover, we will obtain a lower bound for injective dimension of the local cohomology module , in the case where (R,m) is a complete equidimensional local ring.

F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p > 0

The main aim of this article is to study the relation between F-injective singularity and the Frobenius closure of parameter ideals in Noetherian rings of positive characteristic. The paper consists of the following themes, including many other topics.(1)We prove that if every parameter ideal of a Noetherian local ring of prime characteristic p>0 is Frobenius closed, then it is F-injective.(2)We prove a necessary and sufficient condition for the injectivity of the Frobenius action on Hmi(R) for all i≤fm(R), where fm(R) is the finiteness dimension of R. As applications, we prove the following results. (a) If the ring is F-injective, then every ideal generated by a filter regular sequence, whose length is equal to the finiteness dimension of the ring, is Frobenius closed. It is a generalization of a recent result of Ma and which is stated for generalized Cohen–Macaulay local rings. (b) Let (R,m,k) be a generalized Cohen–Macaulay ring of characteristic p>0. If the Frobenius action is injective on the local cohomology Hmi(R) for all i<dim⁡R, then R is Buchsbaum. This gives an answer to a question of Takagi.(3)We consider the problem when the union of two F-injective closed subschemes of a Noetherian Fp-scheme is F-injective. Using this idea, we construct an F-injective local ring R such that R has a parameter ideal that is not Frobenius closed. This result adds a new member to the family of F-singularities.(4)We give the first ideal-theoretic characterization of F-injectivity in terms the Frobenius closure and the limit closure. We also give an answer to the question about when the Frobenius action on the top local cohomology is injective.

On the Auslander–Reiten conjecture for Cohen–Macaulay local rings

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

On the deviation and the type of certain local Cohen–Macaulay rings and numerical semigroups

In J. Herzog and E. Kunz (1973) [6] it was shown that for any pair (d,t)∈N×N+ with (d,t)≠(1,1) there exists a local Cohen–Macaulay ring R having deviation d(R)=d and type t(R)=t. By E. Kunz (1974) [7] the case d(R)=1,t(R)=1 cannot occur. In this paper certain Cohen–Macaulay rings are studied for which there are close relations between deviation, type and embedding dimension. Similar relations for other classes of local rings have been proved in the recent paper by L. Sharifan (2014) [15]. Our relations will be applied to numerical semigroups (or equivalently monomial curves) and lead also to some further cases, generalizing E. Kunz (2016) [8] with ring-theoretic proofs, in which a question of H. Wilf (1978) [16] has a positive answer.

A Northcott type inequality for Buchsbaum-Rim coefficients

In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that, if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

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

ABSTRACTIn 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.

Local Cohomology Annihilators and Macaulayfication

The aim of this paper is to study a deep connection between local cohomology annihilators and Macaulayfication and arithmetic Macaulayfication over a local ring. Local cohomology annihilators appear through the notion of p-standard system of parameters. For a local ring, we prove an equivalence of the existence of Macaulayfications, the existence of a p-standard system of parameters, being a quotient of a Cohen-Macaulay local ring, and the verification of Faltings’ Annihilator theorem. For a finitely generated module which is unmixed and faithful, we prove an equivalence of the existence of an arithmetic Macaulayfication and the existence of a p-standard system of parameters; and both are proved to be equivalent to the existence of an arithmetic Macaulayfication on the ground ring. A connection between Macaulayfication and universal catenaricity is also discussed.

Generalized Hilbert coefficients and Northcott's inequality

Let R be a Cohen–Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread ℓ(I)=d, satisfies the Gd condition and the weak Artin–Nagata property ANd−2−. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)−HI(n), where J is a general minimal reduction of I, PI(n) and HI(n) are respectively the generalized Hilbert–Samuel polynomial and the generalized Hilbert–Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I. As an application, we extend Northcott's inequality to non-m-primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen–Macaulay. We also recover results of G. Colomé-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j1(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.

Remarks on Reflexive Subcategories

In this article, we study reflexive subcategories of the category of finitely generated modules over Cohen–Macaulay local rings R. One of the main results yields a complete classification of the reflexive subcategories for the case when R is a one-dimensional complete local hypersurface of finite Cohen–Macaulay representation type. The other result yields a one-to-one correspondence between the subclasses of a stable category and the reflexive subcategories over a complete local hypersurface. Then, we attempt to extend the result in Takahashi [8, Proposition 4.4] to a reflexive subcategory.

On the intersection of annihilator of the Valabrega–Valla module

Let (A,\mathfrak m) be a Cohen–Macaulay local ring with an infinite residue field and let I be an \mathfrak m -primary ideal. Let \mathbf x = x_1, \ldots, x_r be a A -superficial sequence \wrt \ I . Set \mathcal V_I(\mathbf x) = \bigoplus_{n \geq 1} \frac{I^{n+1} \cap (\mathbf x) }{\mathbf x I^n}. A consequence of a theorem due to Valabrega and Valla is that \mathcal V_I(\mathbf x) = 0 if and only if the initial forms x_1^*, \ldots, x_r^* is a G_I (A) regular sequence. Furthermore this holds if and only if depth G_I(A) \geq r . We show that if depth G_I(A) &lt; r then \mathfrak a_r(I)= \bigcap_{\substack{\mathbf x = x_1, \ldots, x_r \: \mathrm {is \: a}} \: \\ {A-\mathrm {superficial \: sequence \: with \: respect \: to} \: I}} \mathrm {ann}_A \mathcal V_I(\mathbf x) \: \: \: \mathrm {is} \: \mathfrak m \mathrm {-primary}. Suprisingly we also prove that under the same hypotheses, \bigcap_{n \geq 1} \mathfrak a_r(I^n) \: \: \mathrm {is \: also} \: \mathfrak m \mathrm{-primary}.

Perfect linkage of Cohen–Macaulay modules over Cohen–Macaulay rings

In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen–Macaulay modules over Cohen–Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen–Macaulay approximations and Yoshino–Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen–Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of linkage of modules and ideals.

On the vanishing of local cohomology of the absolute integral closure in positive characteristic

The aim of this paper is to extend the main result of C. Huneke and G. Lyubeznik in [Adv. Math. 210 (2007), 498–504] to the class of rings that are images of Cohen–Macaulay local rings. Namely, let R be a local Noetherian domain of positive characteristic that is an image of a Cohen–Macaulay local ring. We prove that all local cohomology of R (below the dimension) maps to zero in a finite extension of the ring. As a direct consequence we obtain that the absolute integral closure of R is a big Cohen–Macaulay algebra. Since every excellent local ring is an image of a Cohen–Macaulay local ring, this result is a generalization of the main result of M. Hochster and Huneke in [Ann. of Math. 135 (1992), 45–79] with a simpler proof.

One-dimensional stable rings

A commutative ring R is stable provided every ideal of R containing a nonzerodivisor is projective as a module over its ring of endomorphisms. The class of stable rings includes the one-dimensional local Cohen–Macaulay rings of multiplicity at most 2, as well as certain rings of higher multiplicity, necessarily analytically ramified. The former are important in the study of modules over Gorenstein rings, while the latter arise in a natural way from generic formal fibers and derivations.We characterize one-dimensional stable local rings in several ways. The characterizations involve the integral closure R‾ of R and the completion of R in a relevant ideal-adic topology. For example, we show: If R is a reduced stable ring, then there are exactly two possibilities for R: (1) R is a Bass ring, that is, R is a reduced Noetherian local ring such that R‾ is finitely generated over R and every ideal of R is generated by two elements; or (2) R is a bad stable domain, that is, R is a one-dimensional stable local domain such that R‾ is not a finitely generated R-module.

Extension Closedness of Syzygies and Local Gorensteinness of Commutative Rings

We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen-Macaulay local ring with canonical module to an arbitrary local ring.

The structure of preenvelopes with respect to maximal Cohen–Macaulay modules

This paper studies the structure of special preenvelopes and envelopes with respect to maximal Cohen–Macaulay modules. We investigate the structure of them in terms of their kernels and cokernels. Moreover, using this result, we also study the structure of special proper coresolutions with respect to maximal Cohen–Macaulay modules over a Henselian Cohen–Macaulay local ring.

Hilbert polynomials of multigraded filtrations of ideals

Hilbert functions and Hilbert polynomials of Zs-graded admissible filtrations of ideals {F(n_)}n_∈Zs such that λ(RF(n_)) is finite for all n_∈Zs are studied. Conditions are provided for the Hilbert function HF(n_):=λ(R/F(n_)) and the corresponding Hilbert polynomial PF(n_) to be equal for all n_∈Ns. A formula for the difference HF(n_)−PF(n_) in terms of local cohomology of the extended Rees algebra of F is proved which is used to obtain sufficient linear relations analogous to the ones given by Huneke and Ooishi among coefficients of PF(n_) so that HF(n_)=PF(n_) for all n_∈Ns. A theorem of Rees about joint reductions of the filtration {IrJs‾}r,s∈Z is generalised for admissible filtrations of ideals in two-dimensional Cohen–Macaulay local rings. Necessary and sufficient conditions are provided for the multi-Rees algebra of an admissible Z2-graded filtration F to be Cohen–Macaulay.

Hilbert coefficients and partial Euler–Poincaré characteristics of Koszul complexes of d-sequences

We construct in a finitely generated module over a Cohen–Macaulay local ring several subquotient modules. In terms of multiplicities of these subquotients, we give precise formulas computing all the partial Euler–Poincaré characteristics of the Koszul complex and the Hilbert coefficients of the module relative to an almost p-standard system of parameters – a very strict subclass of d-sequences on the module. The formulas enable us to establish some comparison between the partial Euler–Poincaré characteristics and the Hilbert coefficients.

On the Canonical Ideals of One-Dimensional Cohen–Macaulay Local Rings

AbstractIn this paper we consider the problem of explicitly finding canonical ideals of one-dimensional Cohen–Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension 2 case, from a Hilbert–Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.

The Dual Hilbert–Samuel Function of a Maximal Cohen-Macaulay Module

Let R be a Cohen–Macaulay local ring with a canonical module ω R . Let I be an 𝔪-primary ideal of R and M, a maximal Cohen–Macaulay R-module. We call the function n⟼ℓ(Hom R (M, ω R /I n+1ω R )) the dual Hilbert–Samuel function of M with respect to I . By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.