arXiv subjects

Alireza Vahidi

Publications and source records attributed to Alireza Vahidi.

10 recordsLinked to original sources

arXiv2025

$(n,d)$-injective and $(n,d)$-flat modules under a special semidualizing bimodule

Let $S$ and $R$ be rings, $n, d\geq 0$ be two integers or $n=\infty$. In this paper, first we introduce special (faithfully) semidualizing bimodule $_S(K_{d-1})_R$, and then introduce and study the concepts of $K_{d-1}$-$(n,d)$-injective (resp. $K_{d-1}$-$(n,d)$-flat) modules as a common generalization of some known modules such as $C$-injective, $C$-weak injective and $C$-$FP_n$-injective (resp. $C$-flat, $C$-weak flat and $C$-$FP_n$-flat) modules. Then we obtain some characterizations of two classes of these modules, namely $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$. We show that the cleasses $\mathcal{I}_{K_{d-1}}^{(n,d)}(R)$ and $\mathcal{F}_{K_{d-1}}^{(n,d)}(S)$ are covering and preenveloping. Also, we investigate Foxby equivalence relative to the classes of this modules. Finally over $n$-coherent rings, we prove that the classes $\mathcal{I}_{ K_{d-1}}^{(n,d)}(R)_{<\infty}$ and $\mathcal{F}_{ K_{d-1}}^{(n,d)}(S)_{<\infty}$ are closed under extentions, kernels of epimorphisms and cokernels of monomorphisms. Keywords: $K_{d-1}$-$(n,d)$-injective module; $K_{d-1}$-$(n,d)$-flat module; Foxby equivalence; special semidualizing bimodule.

math.RA↗

arXiv2024

Cofiniteness and finiteness of associated prime ideals of generalized local cohomology modules

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 associated prime ideals of generalized local cohomology modules. In some cases, we show that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $(\operatorname{FD}_{<n},\mathfrak{a})$-cofinite $R$-module and $\{\mathfrak{p}\in\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,X)):\dim(R/\mathfrak{p})\geq{n}\}$ is a finite set for all $i$. If $R$ is semi-local, we observe that $\operatorname{Ass}_R(\operatorname{H}^{i}_{\mathfrak{a}}(M,N))$ is finite for all $i$ when $\dim_R(M)\leq{3}$ or $\dim_R(N)\leq{3}$. Also, in some situations, we prove that $\operatorname{H}^{i}_{\mathfrak{a}}(M,X)$ is an $\mathfrak{a}$-cofinite $R$-module for all $i$.

math.AC↗

arXiv2024

Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules

Let $R$ and $S$ be rings, $C= {}_SC_R$ a (faithfully) semidualizing bimodule, and $n$ a positive integer or $n=\infty$. In this paper, we introduce the concepts of $C$-$fp_n$-injective $R$-modules and $C$-$fp_n$-flat $S$-modules as a common generalization of some known modules such as $C$-$FP_{n}$-injective (resp. $C$-weak injective) $R$-modules and $C$-$FP_{n}$-flat (resp. $C$-weak flat) $S$-modules. Then we investigate $C$-$fp_{n}$-injective and $C$-$fp_{n}$-flat dimensions of modules, where the classes of these modules, namely $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$, respectively. We study Foxby equivalence relative to these classes, and also the existence of $Cfp_nI(R)_{\leq k}$ and $Cfp_nF(S)_{\leq k}$ preenvelopes and covers. Finally, we study the exchange properties of these classes, as well as preenvelopes (resp. precovers) and Foxby equivalence, under almost excellent extensions of rings.

math.RA↗

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↗

arXiv2018

Extension functors of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$ an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a finitely generated $R$--module $N$, we study the membership of $\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in Serre subcategories of the category of $R$--modules and present some upper bounds for the injective dimension and the Bass numbers of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.

math.AC↗

arXiv2017

A Fractional Gauss-Jacobi quadrature rule for approximating fractional integrals and derivatives

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals.

math.NA↗

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↗

arXiv2011

Lyubeznik numbers of monomial ideals

We study Bass numbers of local cohomology modules supported on squarefree monomial ideals paying special attention to Lyubeznik numbers. We build a dictionary between local cohomology modules and minimal free resolutions that allow us to interpret Lyubeznik numbers as the obstruction to the acyclicity of the linear strands of the Alexander dual ideals. The methods we develop also help us to give a bound for the injective dimension of the local cohomology modules in terms of the dimension of the small support.

math.AC↗

arXiv2010

Torsion functors of local cohomology modules

Through a study of torsion functors of local cohomology modules we improve some non-finiteness results on the top non-zero local cohomology modules with respect to an ideal.

math.AC↗

arXiv2008

Artinian and non-artinian local cohomology modules

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 local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\fa$ such that all local cohomology modules of $M$ with respect to $\fa$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\fq$ of the non-empty set of ideals $\{\fa$ : $\H_\fa^i(M)$ is not artinian for some $i$, $i\geq r$$\}$ is a prime ideal and that all Bass numbers of $\H_\fq^i(M)$ are finite for all $i\geq r$.

math.AC↗