On the dimension of cofinite modules
Let $I$ be an ideal of a commutative Noetherian complete local ring $R$. In the present paper, we establish the equality $\dim R/(I+\Ann_R M)=\dim M$ for all $I$-cofinite $R$-modules $M$.
math.AC↗
arXiv subjects
Majid Rahro Zargar
Publications and source records attributed to Majid Rahro Zargar.
Numerical aspects of complexes of finite homological dimensions
Let $(R,\fm)$ be a local ring, and let $C$ be a semidualizing complex. We establish the equality $r_R(Z) = ν(\Ext^{g-\inf C}_R(Z,C))μ^{\depth C}_R(\mathfrak{m}, C)$ for a homologically finite and bounded complex $Z$ with finite $\GC$-dimension $g$. Additionally, we prove that if $\Ext^i(M,N)=0$ for sufficiently large $i$, while $\id_R\Ext^i(M,N)$ remains finite for all $i$, then both $\pd_R M$ and $\id_R N$ are finite when $M$ and $N$ are finitely generated $R$-modules. These findings extend the recent results of Ghosh and Puthenpurakal \cite{Ghosh}, addressing their questions as presented in \cite[Question 3.9]{Ghosh} and \cite[Question 4.2]{Ghosh}.
math.AC↗
Some characterizations of dualizing complexes in terms of $G_{C}$-dimension
Let $(R,\fm)$ be a local ring and $C$ be a homologically bounded and finitely generated $R$-complex. Then, we prove that $C$ is a dualizing complex of $R$ if and only if $C$ is a Cohen-Macaulay semidualizing complex of type one or $μ_R^{\inf C+\dim_R(C) }(\fm,R)=β_{\inf C}^R(C)$. Also, we show that a semidualizing complex $C$ is dualizing if and only if there exists a type one Cohen-Macaulay $R$-module of finite $G_{C}$-dimension or there exists a type one Cohen-Macaulay $R$-complex of finite $G_{C}$-dimension such that $\dim_R(X)=\dim_R(C)-\gr_C(X)$. Furthermore, for a semidualizing $R$-complex $C$, we prove that $C\sim R$ if and only if there exists a type one Cohen-Macaulay $R$-module $M$ which belongs to the Auslander class $\mathcal{A}_C(R)$.
math.AC↗
Some Characterizations of Relative Sequentially Cohen-Macaulay and Relative Cohen-Macaulay Modules
Let $M$ be an $R$-module over a Noetherian ring $R$ and $\mathfrak{a}$ be an ideal of $R$ with $c={\rm cd}(\mathfrak{a},M)$. First, we prove that $M$ is finite $\mathfrak{a}$-relative Cohen-Macaulay if and only if ${\rm H}_i(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M)))=0$ for all $i\neq c$ and ${\rm H}_c(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^c(M))) \cong \widehat{M}^{\mathfrak{a}}$. Next, over an $\mathfrak{a}$-relative Cohen-Macaulay local ring $(R,\mathfrak{m})$, we provide a characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of $\mathfrak{a}$-relative Cohen-Macaulayness of the $R$-modules ${\rm Ext}^{d-i}_{R}(M,{\rm D}_{\mathfrak{a}})$ for all $i\geq 0$, where ${\rm D}_{\mathfrak{a}} = {\rm Hom}_R({\rm H}^d_{\mathfrak{a}}(R),{\rm E}(R/\mathfrak{m}))$ and $d={\rm cd}(\mathfrak{a},R)$. Finally, we provide another characterization of $\mathfrak{a}$-relative sequentially Cohen-Macaulay modules $M$ in terms of vanishing of the local homology modules ${\rm H}_j(Λ_{\mathfrak{a}}({\rm H}_{\mathfrak{a}}^i(M)))=0$ for all $0\leq i\leq c$ and for all $j\neq i$.
math.AC↗
The derived category analogues of Faltings' Local-global Principle and Annihilator Theorems
Let $\mathcal{Z}$ be a specialization closed subset of $\Spec R$ and $X$ a homologically left-bounded complex with finitely generated homologies. We establish Faltings' Local-global Principle and Annihilator Theorems for the local cohomology modules {$\H_{\mathcal{Z}^i(X).$ }} Our versions contain variations of results already known on these theorems.
math.AC↗
Some duality and equivalence results
Let $(R,\fm)$ be a relative Cohen-Macaulay local ring with respect to an ideal $\fa$ of $R$ and set $c:=\h\fa$. In this paper, we investigate some properties of the Matlis dual $\H_{\fa}^c(R)^{\vee}$ of the $R$-module $\H_{\fa}^c(R)$ and we show that such modules treat like canonical modules over Cohen-Macaulay local rings. Also, we provide some duality and equivalence results with respect to the module $\H_{\fa}^c(R)^{\vee}$ and so these results lead to achieve generalizations of some known results, such as the Local Duality Theorem, which have been provided over a Cohen-Macaulay local ring which admits a canonical module.
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Homological dimensions of rigid 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 dimension of the maximal ideal $\fm$ of $R$ is finite. Furthermore we prove that $R$ must be regular if a single $\Ext_{R}^{n}(I,J)$ vanishes for some integrally closed $\fm$-primary ideals $I$ and $J$ of $R$ and for some integer $n\geq \dim(R)$. Along the way we observe that local rings that admit maximal Cohen-Macaulay Tor-rigid modules are Cohen-Macaulay.
math.AC↗
On flat and Gorenstein flat dimensions of local cohomology modules
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 $\H_{\fa}^i(M)=0$ for all $i\neq n$. It is proved that if $\T(X)<\infty$, then $\T(\H_{\fa}^n(M))\leq\T(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.
math.AC↗
On the relative Cohen-Macaulay modules
Let $R$ be a commutative Noetherian local ring and let $\fa$ be a proper ideal of $R$. A non-zero finitely generated $R$-module $M$ is called relative Cohen-Macaulay with respect to $\fa$ if there is precisely one non vanishing local cohomology modules $\H_{\fa}^{i}(M)$ of $M$. In this paper, as a main result, it is shown that if $M$ is a Gorenstein $R$--module, then $\H_{\fa}^{i}(M)=0$ for all $i\neq c$ where $c=\h_{M}\fa$ is completely encoded in homological properties of $\H_{\fa}^{c}(M)$, in particular in its Bass numbers. Notice that, this result provides a generalization of a result of M. Hellus and P. Schenzel which has been proved before, as a main result, in the case where $M=R$.
math.AC↗
On injective and Gorenstein injective dimensions of local cohomology modules
Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely generated $R$-module $M$ of dimension $d$, it is proved that if $K_{\hat M}$ is Cohen-Macaulay and $\Gid_{R}\H_{\fm}^{d}(M)<\infty$, then$\Gid_{R}\H_{\fm}^{d}(M)=\depth R- d.$ The above results have consequences which improve some known results and provide characterizations of Gorenstein rings.
math.AC↗
Local cohomology modules and Gorenstein injectivity with respect to a semidualizing module
Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently presented.
math.AC↗