Related papers: Cofiniteness over Noetherian complete local rings

Let $R$ denote a commutative Noetherian ring. Brodmann et al. defined and studied the concept of the local-global principle for annihilation of local cohomology modules at level $r\in\mathbb{N}$ for the ideals $\frak a$ and $\frak b$ of…

Let $\frak a$ denote an ideal in a regular local (Noetherian) ring $R$ and let $N$ be a finitely generated $R$-module with support in $V(\frak a)$. The purpose of this paper is to show that all homomorphic images of the $R$-modules…

We obtain various characterizations of commutative Noetherian local rings $(R, \fm)$ in terms of homological dimensions of certain finitely generated modules. For example, we establish that $R$ is Gorenstein if the Gorenstein injective…

Let $(R,m)$ be a local Noetherian ring, let $I\subset R$ be any ideal and let $M$ be a finitely generated $R$-module. In 1990 Craig Huneke conjectured that the local cohomology modules $H^i_I(M)$ have finitely many associated primes for all…

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…

This paper is concerned about the relation between local cohomology modules defined by a pair of ideals and Serre classes of R-modules, as a generalization of results of J. Azami, R. Naghipour and B. Vakili (2009) and M. Asgharzadeh and…

Let $R = \bigoplus_{n \in \mathbb{N}_0} R_n$ be a Noetherian homogeneous ring with local base ring $(R_0, \mathfrak{m}_0)$ and let $M$ and $N$ be finitely generated graded $R$-modules. Let $i,j\in\mathbb{N}_0$. In this paper we will study…

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. In this article we answer affirmatively a question raised by the present author in \cite{B2}. Also, as an immediate consequence of this result it is shown that the…

Let $R$ be a commutative noetherian ring, and $\mathcal{Z}$ a stable under specialization subset of $\Spec(R)$. We introduce a notion of $\mathcal{Z}$-cofiniteness and study its main properties. In the case $\dim(\mathcal{Z})\leq 1$, or…

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$ and let $n$ be a non-negative integer. In this paper, we study $\mathcal{S}_{n}(\mathfrak{a})$, a certain class of $A$-modules and we find some sufficient…

For a Noetherian regular ring $S$ and for a fixed ideal $J\subset S$, assume that the associated primes of local cohomology module $H^i_J(S)$ does not contain $p$ for some $i\geq 0$, and we call this as a property…

Let A be a Noetherian ring. It is shown that any finite A--module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying…

Let $I$ be an ideal of a commutative Noetherian ring $R$. Let $M$ and $N$ be any $R$-modules. We define the generalized completion homology modules $L_i\Lambda^I (N,M)$, for $i\in \mathbb{Z}$, as the homologies of the complex…

Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon \text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let $\mathcal{M}$ be a $F_R$-finite…

Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that…

Our main goal in this paper is to answer new positive cases of the natural generalized version of Hartshorne's celebrated question on cofiniteness of local cohomology modules, and consequently of Huneke's conjecture on the finiteness of…

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated…

Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ of dimension $n$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $H^c_I(M), c = \grade (I,M)$. In particular there is…

Let $(R,\mathfrak{m})$ be a Noetherian regular local ring containing a field of characteristic $p>0$ and $I$ a nonzero ideal of $R$. In this short note, we prove that if $\operatorname{H}^i_I(R)\neq 0$, then $…

Let $(R,\mathfrak m)$ denote an $n$-dimensional complete local Gorenstein ring. For an ideal $I$ of $R$ let $H^i_I(R), i \in \mathbb Z,$ denote the local cohomology modules of $R$ with respect to $I.$ If $H^i_I(R) = 0$ for all $i \not= c =…