Related papers: Artinianness of local cohomology modules of ZD-mod…

Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\fa$ and $\fb$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\fa, \fb$, $\fa\cap\fb$ and $\fa+ \fb$ are studied. When $R$ is…

Let \fa be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions f_{\fa}(M), the finiteness dimension of M with respect to \fa, and, its dual notion q_{\fa}(M), the…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, and let $M$ be a finitely generated $R$-module. For a non-negative integer $t$, we prove that $H_{\fa}^t(M)$ is $\fa$-cofinite whenever $H_{\fa}^t(M)$ is Artinian and…

Let $\fa$ be an ideal of a Noetherian local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. In this paper we introduce some criterions on Artinianness of formal local cohomology, in particular vanishing and finiteness of local…

Let $A$ be a noetherian ring, $\fa$ an ideal of $A$, and $M$ an $A$--module. Some uniform theorems on the artinianness of certain local cohomology modules are proven in a general situation. They generalize and imply previous results about…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. We prove that if $R$ is local with maximal ideal $\fm$ and $ M\otimes_R N$ is of finite…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$ such that $\dim R/\fa=1$ and $M$ a finite $R$--module. We will study cofiniteness and some other properties of the local cohomology modules $\lc^{i}_{\fa}(M)$. For an arbitrary ideal $\fa$…

Let $(R,\mathfrak m)$ be a Noetherian local ring, $I$ an ideal of $R$ and $M$ a weakly finite or a coatomic $R$-module of dimension $n$. In this article, we resolve the Artinianness and non-Artinianness of top local cohomology modules,…

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring, $M$ be a finitely generated $R$-module and $\mathfrak{a}$, $I$ and $J$ be ideals of $R$. We investigate the structure of formal local cohomology modules of…

Let $R$ be a commutative Noetherian ring and $\fa$ an ideal of $R$. We intend to establish the dual of two Faltings' Theorems for local homology modules of an Artinian module. As a consequence of this, we show that, if $A$ is an Artinian…

Let $\fa$ be an ideal of a commutative Noetherian ring $R$ with identity and let $M$ and $N$ be two finitely generated $R$-modules. Let $t$ be a positive integer. It is shown that $\Ass_R(H_{\fa}^t(M,N))$ is contained in the union of the…

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. In this article, we examine the question of whether an arbitrary top local cohomology module, $\operatorname{H}^{\operatorname{cd}(I,M)}_I(M)$, is Artinian, or not.…

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that…

Let $R$ be a commutative noetherian ring, $I,J$ be two ideals of $R$, $M$ be an $R$-module, and $\mathcal{S}$ be a Serre class of $R$-modules. A positive answer to the Huneke$^,$s conjecture is given for a noetherian ring $R$ and minimax…

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $N$ a finite $R$-module. We prove that \cite[Theorem 2.1]{Mel} and \cite[Proposition 3.3 (i)$\Leftrightarrow$(ii)]{B1} are true for any Serre…

Let $R$ be a commutative Noetherian ring with non-zero identity, $\fa$ an ideal of $R$, $M$ a finite $R$--module and $X$ an arbitrary $R$--module. Here, we show that, in the Serre subcategories of the category of $R$--modules, how the…

Let $\fa$ denote an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules with $\pd M< \infty$. It is shown that if $\fa$ is principal or $R$ is complete local and $\fa$ a prime ideal with $\dim…

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, and $M$ an $R$--module. We prove that for a finite module $M$, if $\LC^{i}_{\fa}(M)$ is minimax for all $i\geq r\geq 1$, then $\LC^{i}_{\fa}(M)$ is artinian for $i\geq r$. A Local-global…

Let $\fa$ be an ideal of a local ring $(R,\fm)$ and $M$ a finitely generated $R$-module. We investigate the structure of the formal local cohomology modules ${\vpl}_nH^i_{\fm}(M/\fa^n M)$, $i\geq 0$. We prove several results concerning…

Let $n$ be a non-negative integer, $R$ a commutative Noetherian ring, $\mathfrak{a}$ an ideal of $R$, $M$ and $N$ two finitely generated $R$-modules, and $X$ an arbitrary $R$-module. In this paper, we study cofiniteness and finiteness of…