Searcharxiv⌕ Search

arXiv subjects

Moharram Aghapournahr

Publications and source records attributed to Moharram Aghapournahr.

14 recordsLinked to original sources
arXiv2023

Abelian category of cominimax modules and local cohomology

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, $M$ an arbitrary $R$-module and $X$ a finite $R$-module. We prove that the category of $\fa$-cominimax modules is a Melkersson subcategory of $R$-modules whenever $\dim R\leq 1$ and is an Abelian subcategory whenever $\dim R\leq 2$. We prove a characterization theorem for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(X,M)$ to be $\fa$-cominimax for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$.

math.AC↗
arXiv2023

A characterization result of cofinite local cohomology modules

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 subcategory of $R$-modules. We also prove a characterization theorem for $\lc_{\fa}^{i}(M)$ and $\lc_{\fa}^{i}(N,M)$ to be $\fa$-cofinite for all $i$, whenever one of the following cases holds: (a) $\ara (\fa)\leq 1$, (b) $\dim R/\fa \leq 1$ or (c) $\dim R\leq 2$. In the end we study Artinianness and Artinian $\fa$-cofiniteness of local cohomology modules.

math.AC↗
arXiv2019

Lower bounds of certain general local cohomology modules

Let $R$ be a commutative Noetherian ring, $Φ$ a system of ideals of $R$, $\fa \in Φ$, $M$ an arbitrary $R$-module and $t$ a non-negative integer. Let $\mathcal{S}$ be a Melkersson subcategory of $R$-modules. Among other things, we prove that if $\lc^{i}_Φ(M)$ is in $\mathcal{S}$ for all $i < t$ then $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in Φ$. If $\mathcal{S}$ is the class of $R$-modules $N$ with $\dim N \leq k$ where $k \geq -1$, is an integer, then $\lc^{i}_Φ(M)$ is in $\mathcal{S}$ for all $i < t$ (if and only if) $\lc^{i}_\fa(M)$ is in $\mathcal{S}$ for all $i < t$ and for all $\fa \in Φ$. As consequences we study and compare vanishing, Artinianness and support of general local cohomology and ordinary local cohomology supported at ideals of its system of ideals at initial points $i <t$. We show that $\Supp_{R}(\lc^{\dim M-1}_Φ(M))$ is not necessarily finite whenever $(R,\fm)$ is local and $M$ a finitely generated $R$-module.

math.AC↗
arXiv2019

Finiteness dimensions and cofiniteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ and ideal of $R$, $M$ a finite $R$--module, and $n$ a non-negative integer. In this paper, for an arbitrary $R$--module $X$ which is not necessarily finite, we study the finiteness dimension $f_\mathfrak{a}(M,X)$ and the $n$-th finiteness dimension $f^n_\mathfrak{a}(M,X)$ of $M$ and $X$ with respect to $\mathfrak{a}$. Assume that $\operatorname{Ext}^{i}_{R}(R/\mathfrak{a},X)$ is finite for all $i\leq f^2_\mathfrak{a}(M,X)$ (resp. $i< f^1_\mathfrak{a}(M,X)$). We show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is $\mathfrak{a}$--cofinite for all $i< f^2_\mathfrak{a}(M,X)$ (resp. $i< f^1_\mathfrak{a}(M,X)$) and $\operatorname{Ass}_{R}(\operatorname{H}^{f^2_\mathfrak{a}(M,X)}_{\mathfrak{a}}(M,X))$ (resp. if $\operatorname{Ext}^{f^1_\mathfrak{a}(M,X)}_{R}(R/\mathfrak{a},X)$ is finite, then $\operatorname{Ass}_{R}(\operatorname{H}^{f^1_\mathfrak{a}(M,X)}_{\mathfrak{a}}(M,X))$) is finite.

math.AC↗
arXiv2017

Weakly cofiniteness of local cohomology modules

Let $R$ be a commutative Noetherian ring, $Φ$ a system of ideals of $R$ and $I\in Φ$. Let $M$ be an $R$-module (not necessary $I$-torsion) such that $\dim M\leq 1$, then the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for all $i\geq 0$, if and only if the $R$-module $\Ext^i_{R}(R/I, M)$ is weakly Laskerian, for $i=0, 1$. Let $t\in\Bbb{N}_0$ be an integer and $M$ an $R$-module such that $\Ext^i_R(R/I,M)$ is weakly Laskerian for all $i\leq t+1$. We prove that if the $R$-module $\lc^{i}_Φ(M)$ is ${\rm FD_{\leq 1}}$ for all $i<t$, then $\lc^{i}_Φ(M)$ is $Φ$-weakly cofinite for all $i<t$ and for any ${\rm FD_{\leq 0}}$ (or minimax) submodule $N$ of $\lc^t_Φ(M)$, the $R$-modules $\Hom_R(R/I,\lc^t_Φ(M)/N)$ and $\Ext^1_R(R/I,\lc^t_Φ(M)/N)$ are weakly Laskerian. Let $N$ be a finitely generated $R$-module. We also prove that $\Ext^j_R(N,\lc^{i}_Φ(M))$ and ${\rm Tor}^R_{j}(N,H^{i}_Φ(M))$ are $Φ$-weakly cofinite for all $i$ and $j$ whenever $M$ is weakly Laskerian and $\lc^{i}_Φ(M)$ is ${\rm FD_{\leq 1}}$ for all $i$. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.

math.AC↗
arXiv2013

Some results on generalized local cohomology modules

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 generalized local cohomology modules, the ordinary local cohomology modules and the extension modules behave similarly at the initial points. We conclude some Artinianness and cofiniteness results for $\lc^{n}_{\fa}(M, X)$, and some finiteness results for $\Supp_R(\lc^{n}_{\fa}(M, X))$ and $\Ass_R(\lc^{n}_{\fa}(M, X))$.

math.AC↗
arXiv2012

Cofiniteness of weakly Laskerian local cohomology modules

Let $I$ be an ideal of a Noetherian ring R and M be a finitely generated R-module. We introduce the class of extension modules of finitely generated modules by the class of all modules $T$ with $\dim T\leq n$ and we show it by ${\rm FD_{\leq n}}$ where $n\geq -1$ is an integer. We prove that for any ${\rm FD_{\leq 0}}$(or minimax) submodule N of $H^t_I(M)$ the R-modules ${\rm Hom}_R(R/I,H^{t}_I(M)/N) {\rm and} {\rm Ext}^1_R(R/I,H^{t}_I(M)/N)$ are finitely generated, whenever the modules $H^0_I(M)$, $H^1_I(M)$, ..., $H^{t-1}_I(M)$ are ${\rm FD_{\leq 1}}$ (or weakly Laskerian). As a consequence, it follows that the associated primes of $H^{t}_I(M)/N$ are finite. This generalizes the main results of Bahmanpour and Naghipour, Brodmann and Lashgari, Khashyarmanesh and Salarian, and Hong Quy. We also show that the category $\mathscr {FD}^1(R,I)_{cof}$ of $I$-cofinite ${\rm FD_{\leq1}}$ ~ $R$-modules forms an Abelian subcategory of the category of all $R$-modules.

math.AC↗
arXiv2011

Upper bounds for finiteness of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity and $\fa$ an ideal of $R$. Let $M$ be a finite $R$--module of of finite projective dimension and $N$ an arbitrary finite $R$--module. We characterize the membership of the generalized local cohomology modules $\lc^{i}_{\fa}(M,N)$ in certain Serre subcategories of the category of modules from upper bounds. We define and study the properties of a generalization of cohomological dimension of generalized local cohomology modules. Let $\mathcal S$ be a Serre subcategory of the category of $R$--modules and $n \geqslant \pd M$ be an integer such that $\lc^{i}_{\fa}(M,N)$ belongs to $\mathcal S$ for all $i> n$. If $\fb$ is an ideal of $R$ such that $\lc^{n}_{\fa}(M,N/{\fb}N)$ belongs to $\mathcal S$, It is also shown that the module $\lc^{n}_{\fa}(M,N)/{\fb}\lc^{n}_{\fa}(M,N)$ belongs to $\mathcal S$.

math.AC↗
arXiv2009

On the vanishing and finiteness properties of generalized local cohomology modules

Let $R$ be a commutative noetherian ring, $\fa$ an ideal of $R$ and $M,N$ finite $R$--modules. We prove that the following statements are equivalent. \begin{enumerate} \item[(i)] $\lc^{i}_{\fa}(M,N)$ is finite for all $i< n$. \item[(ii)] $\Coass_R(\lc^{i}_{\fa}(M,N)) \subset \V{(\fa)}$ for all $i< n$. \item[(iii)] $\lc^{i}_{\fa}(M,N)$ is coatomic for all $i< n$. \end{enumerate} If $\pd M$ is finite and $r$ be a non-negative integer such that $r>\pd M$ and $\lc^{i}_{\fa}(M,N)$ is finite (resp. minimax) for all $i\geq r$, then $\lc^{i}_{\fa}(M,N)$ is zero (resp. artinian) for all $i\geq r$.

math.AC↗
arXiv2009

On the vanishing, artinianness and finiteness of local cohomology modules

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 Principle for minimax local cohomology modules is shown. If $\LC^{i}_{\fa}(M)$ is coatomic for $i\leq r$ ($M$ finite) then $\LC^{i}_{\fa}(M)$ is finite for $i\leq r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.

math.AC↗
arXiv2008

Cofiniteness and coassociated primes of local cohomology modules

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 $\fa$ and an $R$--module $M$ (not necessarily finite), we will characterize $\fa$--cofinite artinian local cohomology modules. Certain sets of coassociated primes of top local cohomology modules over local rings are characterized.

math.AC↗
arXiv2008

Artinianness of local cohomology modules

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 the artinianness of some special local cohomology modules in the graded case.

math.AC↗
arXiv2008

Local cohomology: Associated primes, artinianness and asymptotic behaviour

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, $M$ an $R$--module and $n$ a non-negative integer. In this paper we first will study the finiteness properties of the kernel and the cokernel of the natural map $f:\Ext^n_{R}(R/\fa,M)\lo \Hom_{R}(R/\fa,\lc^{n}_{\fa}(M))$. Then we will get some corollaries about the associated primes and artinianness of local cohomology modules. Finally we will study the asymptotic behaviour of the kernel and the cokernel of this natural map in the graded case.

math.AC↗