Annihilation Of Local Cohomology Modules Research Articles

Annihilators of local cohomology modules and restricted flat dimensions

Yoshizawa investigated when local cohomology modules have an annihilator that does not depend on the choice of the defining ideal. In this paper we refine his results and investigate the relationship between annihilators of local cohomology modules and restricted flat dimensions.

Annihilators of local cohomology modules via a classification theorem of the dominant resolving subcategories

This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring.

Annihilators of local cohomology modules over a Cohen-Macaulay ring

A lot of efforts have been put into determining which ideals annihilate local cohomology modules. Faltings' Annihilator Theorem ensured the existence of annihilators of local cohomology modules. In general, however, these annihilators depend on the choice of an ideal. This paper investigates the existence of annihilators of local cohomology modules without dependence on the choice of an ideal. Dao and Takahashi's classification theorem of dominant resolving subcategories helps observe the existence of annihilators over a finite-dimensional Cohen-Macaulay ring.

A short note on annihilators of local cohomology modules

Let [Formula: see text] be a commutative Noetherian domain, [Formula: see text] a nonzero [Formula: see text]-module of finite injective dimension, and [Formula: see text] be a nonzero ideal of [Formula: see text]. In this paper, we prove that whenever [Formula: see text], then the annihilator of [Formula: see text] is zero. Also, we calculate the annihilator of [Formula: see text] for finitely generated [Formula: see text]-modules [Formula: see text] and [Formula: see text] with conditions [Formula: see text] and [Formula: see text]. Moreover, if [Formula: see text] is a regular Noetherian local ring and [Formula: see text] such that [Formula: see text], then we show that there exists an ideal [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text].

Correction to: Annihilators of local cohomology modules and simplicity of rings of differential operators

Correction to: Annihilators of local cohomology modules and simplicity of rings of differential operators

Annihilators of local cohomology modules and simplicity of rings of differential operators

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.

Some results on the cofiniteness and annihilators of local cohomology modules

ABSTRACTLet (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 use of Frobenius map and D-modules in local cohomology

ABSTRACTThe Frobenius depth denoted by F-depth is defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke–Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.

Annihilators of Local Cohomology Modules

Let (R, 𝔪) be a commutative Noetherian complete local ring, M a nonzero finitely generated R-module of dimension n, and I be an ideal of R. In this paper we calculate the annihilator of the top local cohomology module . Also, if (R, 𝔪) is a Noetherian local Cohen–Macaulay ring of dimension d and I is a nonzero proper ideal of R, then we calculate the annihilator of the first nonzero local cohomology module . Finally, we show that if R is an arbitrary Noetherian ring, I an ideal of R, and M is a nonzero finitely generated R-module with cd(I, M) = t ≥ 0, then there exists a submodule N of M such that . This is a generalization of the main result of Bahmanpour, A'zami, and Ghasemi [1] for all ideals of an arbitrary Noetherian ring R.

Non-Cohen-Macaulay Locus and Non Generalized Cohen-Macaulay Locus

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module. In this paper, we improve and extend some known relations among the non-Cohen–Macaulay locus nCM(M), the polynomial type p(M) and the annihilators of local cohomology modules . A characterization for the base ring R being universally catenary with all Cohen–Macaulay formal fibers is given. The nongeneralized Cohen–Macaulay locus nGCM(M) of M is described.

On the Annihilators of Local Cohomology Modules

Let R be a commutative Noetherian ring with nonzero identity, 𝔟a and 𝔟b ideals of R with 𝔟b  ⊆ 𝔟a , and M a finitely generated R-module. In this article, for a non-negative integer n, we show that where is the jth local cohomology module of M with respect to 𝔟b . This implies that there exists a positive integer ℓ, depending on M and 𝔟b , such that for all i < f 𝔟b (M) and all ideals 𝔟a of R with 𝔟b  ⊆ 𝔟a where f 𝔟b (M) is the finiteness dimension of M relative to 𝔟b . Also, we obtain a new characterization of the concept of M-grade of an ideal of R.

A new depth and the annihilation of local cohomology modules

Let J\(\subseteq\)I be two proper ideals of a commutative Noetherian ring and M a finitely generated module. Strong relative depth is defined and characterized. It is proved that this depth is just the maximum integer n such that \(H^i_I(M)\) can be annihilated by some power of J for all i ≤ n. It turns out that the local-global principle for the annihilation of local cohomology modules can be formulated as a natural property of this new depth.

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.

A new version of Local-Global Principle for annihilations of local cohomology modules

Let $R$ be a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $N$ be a finitely generated $R$-module. We introduce a generalization of the $\mathfrak b$-finiteness dimension of $f^{\mathfrak b}_{\mathfrak a}(N)$ re

On annihilators and associated primes of local cohomology modules

We establish the Local-global Principle for the annihilation of local cohomology modules over an arbitrary commutative Noetherian ring R at level 2. We also establish the same principle at all levels over an arbitrary commutative Noetherian ring of dimension not exceeding 4. We explore interrelations between the principle and the Annihilator Theorem for local cohomology, and show that, if R is universally catenary and all formal fibres of all localizations of R satisfy Serre's condition ( S r ), then the Annihilator Theorem for local cohomology holds at level r over R if and only if the Local-global Principle for the annihilation of local cohomology modules holds at level r over R. Moreover, we show that certain local cohomology modules have only finitely many associated primes. This provides motivation for the study of conditions under which the set ⋃ m,n∈ N Ass(M/(x m,y n)M) (where M is a finitely generated R-module and x, y∈ R) is finite: an example due to M. Katzman shows that this set is not always finite; we provide some sufficient conditions for its finiteness.

Uniform annihilation of local cohomology and of Koszul homology

Let R be a ring (all rings considered here are commutative with identity and Noetherian), M a finitely generated R-module, and I an ideal of R. The jth local cohomology module of M with support in I is defined byIn this paper, we prove a uniform version of a theorem of Brodmann about annihilation of local cohomology modules. As a corollary of this, we deduce a generalization of a theorem of Hochster and Huneke about uniform annihilation of Koszul homology.