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.     

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.

     

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.     

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．     

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.     

Faltings' theorem for the annihilation of local cohomology modules over a Gorenstein ring

In this paper we study the Annihilator Theorem and the Local-global Principle for the annihilation of local cohomology modules over a (not necessarily finite-dimensional) Noetherian Gorenstein ring.

     

Artinianess of graded local cohomology modules

Let be a Noetherian homogeneous ring with local base ring and let be a finitely generated graded -module. Let be the largest integer such that is not Artinian. We will prove that are Artinian for all and there exists a polynomial of degree less than such that for all . Let be the first integer such that the local cohomology module is not cofinite. We will show that for all the graded module is Artinian.