Cohen-Macaulay Ring Research Articles

A FEW COMMENTS ON MATLIS DUALITY

For a Noetherian local ring (R, m) with p ∈ Spec(R), we denote the R-injective hull of R/p by ER(R/p). We show that it has an Rˆp -module structure, and there is an isomorphism ER(R/p) ∼= ERˆp (Rˆp/pRˆp ), where Rˆp stands for the p-adic completion of R. Moreover, for a complete Cohen-Macaulay ring R, the module D(ER(R/p)) is isomorphic to Rˆp provided that dim(R/p) = 1, where D(·) denotes the Matlis dual functor HomR(·, ER(R/m)). Here, Rˆp denotes the completion of Rp with respect to the maximal ideal pRp. These results extend those of Matlis (see [11]) shown in the case of the maximal ideal m.

Well-Covered and Cohen–Macaulay Unitary Cayley Graphs

Let G(R) be the unitary Cayley graph corresponding to a finite commutative ring R with nonzero identity. Let ΔG(R) be the simplicial complex associated to G(R), whose faces correspond to the independent sets of G(R). We study well-coverednees of G(R) and Cohen–Macaulayness of ΔG(R), i.e., its Stanley–Reisner ring k [ΔG(R)] is a Cohen–Macaulay ring. Furthermore, we show that a unitary Cayley graph is shellable and Gorenstein if it is 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.

Cohen–Macaulay Circulant Graphs

Let G be the circulant graph C n (S) with , and let I(G) denote the edge ideal in the ring R = k[x 1,…, x n ]. We consider the problem of determining when G is Cohen–Macaulay, i.e, R/I(G) is a Cohen–Macaulay ring. Because a Cohen–Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C n (1, 2,…, d). We also characterize which cubic circulant graphs are Cohen–Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen–Macaulay property.

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.

Computing families of Cohen-Macaulay and Gorenstein rings

We characterize Cohen-Macaulay and Gorenstein rings obtained from certain types of convex body semigroups. Algorithmic methods to check if a polygonal or circle semigroup is Cohen-Macaulay/Gorenstein are given. We also provide some families of Cohen-Macaulay and Gorenstein rings.

Multiplicities of Semidualizing Modules

A finitely generated module C over a commutative noetherian ring R is semidualizing if Hom R(C, C) ≅ R and for all i ≥ 1. For certain local Cohen–Macaulay rings (R, 𝔪), we verify the equality of Hilbert-Samuel multiplicities eR(J; C) = eR(J; R) for all semidualizing R-modules C and all 𝔪-primary ideals J. The classes of rings we investigate include those that are determined by ideals defining fat point schemes in projective space or by monomial ideals.

Stability of Restricted Homological Dimensions

In this note, the stability of restricted homological dimensions of complexes over commutative noetherian local ring is discussed. For instance, letRbe a Cohen-Macaulay ring andX, Y. IfXis bounded below with Gpd,Yis bounded above and =0 for all , then Rfd=Rfd+Rfd.

Super-stretched and graded countable Cohen–Macaulay type

We define what it means for a Cohen–Macaulay ring to be super-stretched and show that Cohen–Macaulay rings of graded countable Cohen–Macaulay type are super-stretched. We use this result to show that rings of graded countable Cohen–Macaulay type, and positive dimension, have possible h-vectors (1), (1,n), or (1,n,1). Further, one-dimensional standard graded Gorenstein rings of graded countable type are shown to be hypersurfaces; this result is not known in higher dimensions. In the non-Gorenstein case, rings of graded countable Cohen–Macaulay type of dimension larger than 2 are shown to be of minimal multiplicity.

K-groups for rings of finite Cohen–Macaulay type

Abstract For a local Cohen–Macaulay ring R of finite CM-type, Yoshino has applied methods of Auslander and Reiten to compute the Grothendieck group K0 of the category mod R of finitely generated R-modules. For the same type of rings, we compute in this paper the first Quillen K-group K1(mod R). We also describe the group homomorphism R* → K1(mod R) induced by the inclusion functor proj R → mod R and illustrate our results with concrete examples.

The Joint Reduction Number and Upper Bounds of Hilbert Series of Fiber Cones

Let (R,𝔪) be a Cohen-Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. Let a1,…,ad be a joint reduction of (I[d-1]|K), and set L=(a1,…,ad), J=(a1,…,ad-1). When depth G(I) ≥ d-1 and depth FK(I) ≥ d-2, we show that the lengths [Formula: see text], [Formula: see text] and the joint reduction number rL(I|K) are independent of L. In the general case, we give an upper bound of the Hilbert series of FK(I). When depth G(I) ≥ d-1, we also provide a characterization, in terms of the Hilbert series of FK(I), of the condition depth FK(I) ≥ d-1.

Vanishing of Ext, cluster tilting modules and finite global dimension of endomorphism rings

Let $R$ be a Cohen-Macaulay ring and $M$ a maximal Cohen-Macaulay $R$-module. Inspired by recent striking work by Iyama, Burban-Iyama-Keller-Reiten and van den Bergh we study the question of when the endomorphism ring of $M$ has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension version of Auslander correspondence and existence of non-commutative crepant resolutions. We also recover and extend to positive characteristics a recent Theorem by Burban-Iyama-Keller-Reiten on cluster-tilting objects in the category of maximal Cohen-Macaulay modules over reduced $1$-dimensional hypersurfaces.

On Restricted Injective Dimensions of Complexes

Let R be a commutative Noetherian ring. If cmd R ≤ 1, then it is shown that for any R-module the large restricted injective dimension is a refinement of the Gorenstein injective dimension. If (R, 𝔪) is a Cohen–Macaulay local ring and , then it is proven that the large restricted injective dimension satisfies a Bass formula, Rid R Y = depth R − width R Y. In addition, a characterization of almost Cohen–Macaulay ring is obtained.

The equality [formula omitted] in sequentially Cohen–Macaulay rings

Let R be a sequentially Cohen–Macaulay local ring and assume that dimR⩾2 or that dimR=1 and e(R)>1. Then the equality I2=qI holds true for every good parameter ideal q of R contained in a sufficiently high power of the maximal ideal m, where I=q:Rm. The structure of the graded rings gr(I)=⊕n⩾0In/In+1, R(I)=⊕n⩾0In, and R′(I)=⊕n∈ZIn associated to I is explored in connection to their sequential Cohen–Macaulayness.

On the noncommutative Bondal–Orlov conjecture

Let R be a normal, equi-codimensional Cohen–Macaulay ring of dimension with a canonical module . We give a sufficient criterion that establishes a derived equivalence between the noncommutative crepant resolutions of R . When , this criterion is always satisfied and so all noncommutative crepant resolutions of R are derived equivalent. Our method is based on cluster tilting theory for commutative algebras, developed by Iyama and Wemyss (2010).

Grade Filtration of Linear Functional Systems

The grade (purity) filtration of a finitely generated left module M over an Auslander regular ring D is a built-in classification of the elements of M in terms of their grades (or their (co)dimensions if D is also a Cohen-Macaulay ring). In this paper, we show how grade filtration can be explicitly characterized by means of elementary methods of homological algebra. Our approach avoids using sophisticated methods such as bidualizing complexes, spectral sequences, associated cohomology, or Spencer cohomology used in the literature of algebraic analysis. Efficient implementations dedicated to the computation of grade filtration can then be easily developed in the standard computer algebra systems. Moreover, this characterization of grade filtration is shown to induce a new presentation of the left D-module M which is defined by a block-triangular matrix formed by equidimensional diagonal blocks. The linear functional system associated with the left D-module M can then be integrated in cascade by successively solving inhomogeneous linear functional systems defined by equidimensional homogeneous linear systems of increasing dimension. This equivalent linear system generally simplifies the computation of closed-form solutions of the original linear system. In particular, many classes of underdetermined/overdetermined linear systems of partial differential equations can be explicitly integrated by the Maple package PurityFiltration and the GAP package homalg, but not by the standard PDE solvers of computer algebra systems such as Maple.

Residual intersections of Licci ideals are Glicci

This paper deals with the interplay of two generalizations of classical complete intersection linkage, namely Gorenstein linkage and residual intersection. Recall that two proper ideals I and J in a Cohen-Macaulay ring R are said to be linked or linked with respect to a, written I ∼ J, if J = a : I and I = a : J for some complete intersection ideal a. The link is called geometric in case I + J has height at least g + 1, where g = ht a. As is well known, two linked ideals are automatically unmixed of height g, and conversely, whenever a ( I are ideals of height g in a Gorenstein ring with a a complete intersection and I unmixed, then I ∼ a : I is a link [32] (see also [15]). A sequence of two links I ∼ J ∼ K is referred to as a double link. In a similar manner we define the (even, odd) linkage class of an ideal I, which is the set of all ideals obtained from I by a finite (even, odd) number of links. Finally, one says that an ideal is licci if it belongs to the linkage class of a complete intersection. Licci ideals have been studied extensively. They are known to be perfect for instance, and hence define Cohen-Macaulay rings [32]. In addition they share more subtle homological properties of complete intersections: Licci ideals are strongly nonobstructed (provided R is Gorenstein) [5], are strongly Cohen-Macaulay [21], and the shifts in their homogeneous minimal free resolution grow ‘sufficiently fast’ (provided R is a polynomial ring over a field and the ideal is homogeneous) [24]. These more refined properties have often been used to verify that a given ideal fails to be licci, but they also suggest that the classification provided by complete intersection linkage might be too fine for some purposes. Hence in recent years the emphasis has shifted to the more inclusive notion of Gorenstein linkage, where the complete intersection ideal a in the definition of linkage is replaced by an unmixed Gorenstein ideal a, meaning an unmixed ideal so that R/a is Gorenstein. The first systematic study of this notion can be found in [34]. Again one talks about Gorenstein double linkage, the (even, odd) Gorenstein linkage class of an ideal, and the property of being glicci, which means that an ideal belongs to the Gorenstein linkage class of a complete intersection. The same remarks as above apply

On the Apéry sets of monomial curves

In this paper, we use the Apéry table of the numerical semigroup associated to an affine monomial curve in order to characterize arithmetic properties and invariants of its tangent cone. In particular, we precise the shape of the Apéry table of a numerical semigroup of embedding dimension 3, when the tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new proofs for two conjectures raised by Sapko (Commun. Algebra 29:4759–4773, 2001) and Shen (Commun. Algebra 39:1922–1940, 2001). We also provide a new simple proof in the case of monomial curves for Sally’s conjecture (Numbers of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a one-dimensional Cohen-Macaulay ring with embedding dimension three is non-decreasing. Finally, we obtain that monomial curves of embedding dimension 4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert functions. Numerous examples are provided to illustrate the results, most of them computed by using the NumericalSgps package of GAP (Delgado et al., NumericalSgps-a GAP package, 2006).

Cohen–Macaulay approximation in fibred categories

We extend the Auslander–Buchweitz axioms and prove Cohen–Macaulay approximation results for fibred categories. We show that these axioms apply for the fibred category of pairs consisting of a finite type flat family of Cohen–Macaulay rings and modules. In particular such a pair admits an approximation with a flat family of maximal Cohen–Macaulay modules and a hull with a flat family of modules with finite injective dimension. The existence of minimal approximations and hulls in the local, flat case implies extension of upper semi-continuous invariants. As an example of MCM approximation we define a relative version of Auslanderʼs fundamental module.

Frobenius criteria of freeness and Gorensteinness

Let F n(−) be the Frobenius functor of Peskine and Szpiro. In this note, we show that the maximal Cohen–Macaulayness of F n(M) forces M to be free, provided M has a rank. We apply this result to obtain several Frobenius-related criteria for the Gorensteinness of a local ring R, one of which improves a previous characterization due to Hanes and Huneke. We also establish a special class of finite length modules over Cohen–Macaulay rings, which are rigid against Frobenius.