Related papers: The index of reducibility of parameter ideals in l…

We prove that if M is a finitely-generated module of dimension d with finite local cohomologies over a Noetherian local ring, and if the ith local cohomology module of M is zero unless i = d, i = 0, and i = r for some r strictly between 0…

Let $(R, \frak m)$ be a Noetherian local ring, $M$ a finitely generated $R$-module. The aim of this paper is to prove a uniform formula for the index of reducibility of paprameter ideals of $M$ provided the polynomial type of $M$ is at most…

Let $(R, \frak m)$ be a Noetherian local ring and $M$ a finitely generated $R$-module of dimension $d$. A famous result of Northcott says that if $M$ is Cohen-Macaulay, then the index of reducibility of parameter ideals on $M$ is an…

Let $R$ be a noetherian local ring. We consider the following quastion: Does there exist an integer $n$ such that all idelas generated by a system of parameters contained in the $n$-th power of the maximal ideal have the same Betti numbers?…

Let $M$ be a finitely generated module over a Noetherian ring $R$ and $N$ a submodule. The index of reducibility ir$_M(N)$ is the number of irreducible submodules that appear in an irredundant irreducible decomposition of $N$ (this number…

A commutative noetherian local ring $(R,\mathfrak{m})$ is Gorenstein if and only if every parameter ideal of $R$ is irreducible. Although irreducible parameter ideals may exist in non-Gorenstein rings, Marley, Rogers, and Sakurai show there…

Let (R,m) be a complete local ring, a an ideal of R and M a finitely generated R-module. The aim of this paper is to show that for any non-negative integer n, the least integer i such that the i-th local cohomology with respect to a is not…

The purpose of this paper is to give affirmative answers to two open questions as follows. Let $(R, \m)$ be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers \cite {R} and the…

Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$ and $M, N$ two finitely generated $R$-modules. By using a spectral sequence argument, it is shown that if either $\mathrm{dim}_RM\leq2$ and $\mathrm{H}^{i}_\mathfrak{a}(N)$…

Let $\mathfrak{a}$ be an ideal of a noetherian (not necessarily local) ring $R$ and $M$ an $R$-module with $\mathrm{Supp}_RM\subseteq\mathrm{V}(\mathfrak{a})$. We show that if $\mathrm{dim}_RM\leq2$, then $M$ is $\mathfrak{a}$-cofinite if…

Let $R$ denote a commutative Noetherian (not necessarily local) ring, $M$ an arbitrary $R$-module and $I$ an ideal of $R$ of dimension one. It is shown that the $R$-module $\Ext^i_R(R/I,M)$ is finitely generated (resp. weakly Laskerian) for…

Let $(R,\fr m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M, N$ two finitely generated $R$-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that…

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the…

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with $\dim_RN\leq k$, where $k$ is an integer. It is proved that $\inf\{i:…

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)]…

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$, $N$ two finitely generated $R$-modules. The aim of this paper is to investigate the $I$-cofiniteness of generalized local cohomology modules $\displaystyle…

Let $(R,\mathfrak{m})$ be a local Noetherian ring with residue field $k$. While much is known about the generating sets of reductions of ideals of $R$ if $k$ is infinite, the case in which $k$ is finite is less well understood. We…

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of…

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module with $\operatorname{cd}(I,M)=c$. In this article, we first show that there exists a descending chain of ideals $I=I_c\supsetneq I_{c-1}\supsetneq \cdots \supsetneq I_0$…

For a noetherian ring R we call an R-module M cofinite if there exists an ideal I of R such that M is I-cofinite; we show that every cofinite module M satisfies dim_R(M)<=injdimR(M). As an application we study the question which local…