Integral Closures of Cohen-Macaulay Monomial Ideals

ABSTRACT

The purpose of this paper is to present a family of Cohen-Macaulay monomial ideals such that their integral closures have embedded components and hence are not Cohen-Macaulay.     

The essential ideal is a Cohen-Macaulay module

Let be a finite -group which does not contain a rank two elementary abelian -group as a direct factor. Then the ideal of essential classes in the mod- cohomology ring of is a Cohen-Macaulay module whose Krull dimension is the -rank of the centre of . This basically answers in the affirmative a question posed by J. F. Carlson (Question 5.4 in Problems in the calculation of group cohomology, 1999).

     

The Hilbert function of a maximal Cohen-Macaulay module

We study Hilbert functions of maximal CM modules over CM local rings. When A is a hypersurface ring with dimension d>0, we show that the Hilbert function of M with respect to is non-decreasing. If A=Q/(f) for some regular local ring Q, we determine a lower bound for e₀(M) and e₁(M) and analyze the case when equality holds. When A is Gorenstein a relation between the second Hilbert coefficient of M, A and S^A(M)= (Syz^A₁(M*))* is found when G(M) is CM and depthG(A)≥d−1. We give bounds for the first Hilbert coefficients of the canonical module of a CM local ring and analyze when equality holds. We also give good bounds on Hilbert coefficients of M when M is maximal CM and G(M) is CM.     

Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring R with respect to a semidualzing module C, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces C to be trivial and R to be Gorenstein.     

Ratliff-Rush closures of ideals with respect to a Noetherian module

Let R be a commutative Noetherian ring, E a non-zero finitely generated R-module and I a E-proper ideal of R. The purpose of this paper is to provide some new characterizations of when all powers of I are Ratliff-Rush closed with respect to E and to answer a question raised by W. Heinzer et al. in (The Ratliff-Rush Ideals in a Noetherian Ring: A Survey, in Methods in Module Theory, Dekker, New York, 1992, pp. 149-159).     

Strongly Cohen-Macaulay Ideals of Small Second Analytic Deviation

In this paper, the strongly Cohen-Macaulay ideals of secondanalytic deviation one are characterized in terms of the depthproperties of the powers of the ideal in the ‘standardrange’. This provides an explanation of the behaviourof certain ideals that have appeared in the literature. 2000Mathematics Subject Classification 13H10 (primary), 13C40, 13D02,13D25 (secondary).     

Relative Gorenstein injective covers with respect to a semidualizing module

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to C which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every G _C -injective module G, the character module G ⁺ is G _C -flat, then the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class \(\mathcal{GI}_{C}(R)\cap\mathcal{A}_C(R)\) is covering.     

On the deviation and the type of a Cohen-Macaulay ring

In this note we discuss the question, which pairs of integers (d,r) can occur as the deviation d and the type r of a Cohen-Macaulay domain.     

Modules of finite homological dimension with respect to a semidualizing module

We prove versions of results of Foxby and Holm about modules of finite (Gorenstein) injective dimension and finite (Gorenstein) projective dimension with respect to a semidualizing module. We also verify special cases of a question of Takahashi and White. This research was conducted while S.S.-W. visited the IPM in Tehran during July 2008. The research of S.Y. was supported in part by a grant from the IPM (No. 87130211).     

Extremal Betti Numbers of Some Cohen-Macaulay Binomial Edge Ideals

We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones,and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals:Cohen-Macaulay bipartite and fan graphs.In addition,we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.     

Distribution of points of one-dimensional quasilattices with respect to a variable module

We consider one-dimensional quasiperiodic Fibonacci tilings. Namely, we study sets of vertices of these tilings that represent one-dimensional quasilattices defined on the base of a parameterization by rotations of a circle, and the distribution of points of quasilattices with respect to a variable module. We show that the distribution with respect to some modules is not uniform. We describe the distribution function and its integral representation, and estimate the remainder in the problem of the distribution of points of a quasilattice for corresponding modules.     

Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let ${(R,\mathfrak{m})}$ be a local ring, and let C be a semidualizing R-module. In this paper, we are concerned with the C-injective and G _C -injective dimensions of certain local cohomology modules of R. Firstly, the injective dimension of C and the above quantities are compared. Secondly, as an application of the above comparisons, a characterization of a dualizing module of R is given. Finally, it is shown that if R is Cohen-Macaulay of dimension d such that ${\rm H}_{\mathfrak{m}}^{d}(C)$ is C-injective, then R is Gorenstein. This is an answer to a question which was recently raised.     

On the endomorphism ring of a module noetherian with respect to a torsion theory

IfM is a module torsionfree and noetherian with respect to a torsion theory, ifS is the endomorphism ring ofM, and ifL(S) is the ideal ofS consisting of all endomorphisms with large kernels, thenL(S) is nilpotent and a bound on the index of nilpotency ofL(S) is given.     

Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

Let ${(R, \mathfrak{m})}$ be a commutative Noetherian local ring of Krull dimension d, and let C be a semidualizing R-module. In this paper, it is shown that if R is complete, then C is a dualizing module if and only if the top local cohomology module of ${R, H _{\mathfrak{m}} ^{d} (R)}$ , has finite G _C -injective dimension. This generalizes a recent result due to Yoshizawa, where the ring is assumed to be complete Cohen-Macaulay.     

On measures ergodic with respect to an analytic equivalence relation

In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.

     

Generalized functions which are analytic with respect to some of their variables

For a generalized function which is analytic with respect to some of its variables in the sense of Gording and Malgrange we prove a theorem on representation in the form of a power series in these variables which converges in D′ and a corresponding uniqueness theorem. We also obtain necessary and sufficient conditions for a generalized function to be entire with respect to some of its variables.     

Erratum to: Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module

This note makes a correction to the paper “Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module”.     

Subordination and superordination results for analytic functions with respect to symmetrical points

The theory of symmetric functions has many applications in the investigation of fixed points, estimation of absolute values of some integrals and obtaining the results of the type of Cartan’s uniqueness theorem. In this paper, we solve some differential subordinations and superordinations involving analytic functions with respect to the symmetric points and also derive some sandwich results under certain assumptions on the parameters involved. The various results presented in this paper are shown to apply to yield the corresponding (new or known) results for many simpler function classes.