A companion ideal of a multiplication module

Let M _R be a faithful multiplication module, where R is a commutative ring. As defined by Anderson, this ideal has proved to be useful in studying multiplication modules. First of all a cancellation law involving M and the ideals contained in is proved. Among various applications given, the following result is proved:: There exists a canonical isomorphism from onto such that for any ( Hom _R(M,M), x ( M, a ( (M), (xa) = x.(()(a). As an application of this later result it is proved that M is quasi-injective if and only if (M) is quasi-injective.     

On coLaskerian module and local cohomology

The notion of generalized Laskerian modules was introduced recently by the authors. In this paper, we continue our investigation of such modules. Moreover, we are going to generalize some results on finiteness of associated primes of local cohomology modules to the category of generalized Laskerian modules.     

关于广义分次局部上同调的tame性的一个结果

设R=＋n∈N0Rn（R=R0[R1]）是分次Noether交换环,（R0,m0）是一个局部环,R＋=＋n∈NRn;设N是一个有限生成Z-分次R-模,这里N、N0、Z分别表示全体正整数、全体非负整数和全体格致所构成的集合．令h=sup{i∈Z｜HR＋^i（N）不是Artin模}．Dibaei和Nazari证明了HR＋^h（N）是tame模．我们将该结果推广到了广义分次局部上同调模的情形．     

局部上同调模的Artin性质

In this paper,let (R,m)be a Noetherian local ring,I(?)R an ideal, M and N be two finitely generated modules.Firstly,we study the properties of H_I~t(M),t=f-depth(I,M) and discuss the relationship between the Artinianness of H_I~i(M,N) and the Artinianness of H_I~i(N).Then,we get that H_I~d(M,N)is I-cofinite,if(R,m)is a d-dimensional Gorenstein local ring.     

A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

Bounds for regularity and coregularity of graded modules

Let M be a finitely generated graded module over a Noetherian homogeneous ring R with local base ring (R ₀, m₀). If R ₀ is of dimension one, then we show that reg ⁱ⁺¹(M) and coreg ⁱ⁺¹(M) are bounded for all i ∈ ℕ₀. We improve these bounds, if in addition, R ₀ is either regular or analytically irreducible of unequal characteristic.     

Duplication of a module along an ideal

Let A be a commutative ring and I an ideal of A. The amalgamated duplication of A along I, denoted by \({A \bowtie I}\) , is the special subring of \({A \times A}\) defined by \({A \bowtie I } := \pi \times_{\frac{A}{I}} \pi = \{(a, a + i) \mid a \in A, i \in I\}\) . We are interested in some basic and homological properties of a special kind of \({A \bowtie I}\) -modules, called the duplication of M along I with M is an A-module, and defined by \({M \bowtie I := \{(m, m') \in M \times M \mid m - m^{\prime} \in IM\}}\) . The new results generalize some results on amalgamated duplication of a ring along an ideal.     

Asymptotic Behaviour and Artinian Property of Graded Local Cohomology Modules

In this article, considering the difference between the finiteness dimension and cohomological dimension of a finitely generated graded module, we investigate the asymptotic behaviour of grades of components of a graded local cohomology module with respect to the irrelevant ideal. To this end, we study some Artinian and tameness properties of certain graded local cohomology modules.     

Cohomology theories based on Gorenstein injective modules

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.

     

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).

     

Prime Submodules and a Sheaf on the Prime Spectra of Modules

We define and investigate a sheaf of modules on the prime spectra of modules, and it is shown that there is an isomorphism between the sections of this sheaf and the ideal transform module.     

Some results on the cofiniteness and annihilators of local cohomology modules

Let (R,𝔪) be a Noetherian local ring. In this paper, we give a new characterization for the universal catenaricity of R and the Cohen–Macaulayness of all its formal fibers, and we study the cofiniteness and annihilators of local cohomology modules.     

A Note on the Associated Primes of Local Cohomology Modules

In this note we give a simple proof of the following result: Let R be a commutative Noetherian ring,  an ideal of R and M a finite R-module, if H_ ⁱ (M) has finite support for all i < n, then Ass(H_ ⁿ (M)) is finite.