arXiv subjects

Reza Sazeedeh

Publications and source records attributed to Reza Sazeedeh.

At least 19 recordsLinked to original sources

arXiv2026

Classifying subcategories of a Grothendieck category via its spectral category

Let $\mathcal{A}$ be a Grothendieck category. In this paper we classify subcategories of $\mathcal{A}$ via a support notion defined by Krause \cite{Kr2024} based on the spectral category of $\mathcal{A}$. We show that this aligns with Nemman's support \cite{Ne1992}, and thereby extends earlier classifications of subcategories of the category of modules over a commutative noetherian ring and subcategories of its derived category. Over a locally coherent category $\mathcal{A}$, we show that this support agrees with open subsets of the Ziegler topology on $\mathcal{A}$.

math.RT↗

arXiv2025

Classifying localizing subcategories of a locally coherent category

Let $\cA$ be a locally coherent Grothendieck category, $\fp\cA$ be the full subcategory of $\cA$ consisting of finitely presented objects and $\ASpec\cA$ be the atom spectrum of $\cA$. In this paper, we classify localizing subcategories of finite type of $\cA$ via open subsets of $\ASpec\cA$. We investigate $\ASpec\fp\cA$ and show that if $\ASpec\fp\cA=\ASpec\cA$, then $\cA$ is locally noetherian. As an application, we specialize our investigation to the case of commutative coherent rings.

math.CT↗

arXiv2025

On the category of cofinite complexes and modules

Let $A$ be a commutative noetherian ring, let $\mathfrak a$ be an ideal of $A$. In this paper, we extend Hartshorne's characterization of cofinite complexes to more general classes of rings. We also determine conditions under which Hartshorne's fourth question [H1] admits an affirmative answer. Finally, we investigate the cofiniteness of complexes of $\frak a$-cofinite modules for rings of lower dimensions.

math.AC↗

arXiv2025

Gorenstein injective filtrations over rings with dualizing complexes

Let $R$ be a commutative noetherian ring. Enochs and Huang [EH] proved that over a Gorenstein ring of Krull dimension $d$, every Gorenstein injective module admits a finite filtration of Gorenstein injective submodules. In this paper, we extend this result to rings admitting a dualizing complex and we provide such filtrations using Auslander categories and section functors.

math.AC↗

arXiv2025

Spectrum of an abelian category via premonoform objects

Let $\cA$ be an abelian category. In this paper, we study ${\rm (n)PSpec}\cA$, a topological space formed by equivalence classes derived from an equivalence relation on (noetherian) premonoform objects. We classify torsion classes of $\cA$ via closed subclasses of $\nPSpec\cA$. We introduce a new topology on $\PSpec\cA$ and we classify Serre subcategories of $\noeth A$ and localizing subcategories of $\cA$ using this topology. If $A$ is a commutative noetherian ring, we show that $\nPSpec A$ is homeomorphic to $\Spec A$. Moreover, there is a one-to-one correspondence between the closed subsets of $\nPSpec A$ and the open subsets of $\ASpec A$, the atom spectrum of $A$. Finally, we explore the relationships between the new subctegories of $\Mod A$ and subsets of $\nPSpec A$ introduced in this paper, and the known subcategories of $\Mod A$ and subsets of other spectra of $A$.

math.CT↗

arXiv2024

Bass numbers and endomorphism rings of Gorenstein injective modules

Let $R$ be a commutative noetherian ring admitting a dualizing complex and let $\mathfrak p$ be a prime ideal of $R$. In this paper we investigate when $G(R/\frak p)$ is an $R_{\frak p}$-module. We give some necessary and sufficient conditions under which $G(R/\frak p)$ is an $R_{\frak p}$-module. We also study the Bass numbers of $G(R/\frak p)$ and we show that if ${\rm Gid}_RR/\frak p$ is finite, then $μ^i(\frak q,G(R/\frak p))$ is finite for all $i\geq 0$ and all $\frak q\in{\rm Spec} R$. If ${\rm Gpd}_RR/\frak p$ is finite, then $μ^i(\frak p,G(R/\frak p))$ is finite for all $i\geq 0$. We define a subring $S(\frak p)_{\frak p}$ of ${\rm End}_{R_{\frak p}}(G(R_{\frak p}/\frak pR_{\frak p}))$ and we show that it is noetherian and contains a subring which is a quotient of $\widehat{R_{\frak p}}$.

math.AC↗

arXiv2023

Cofiniteness of modules and local cohomology

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 conditions so that a module belongs to $\mathcal{S}_{n}(\mathfrak{a})$. Moreover, we study the cofiniteness of local cohomology modules when $\dim A/\frak a\geq 3$.

math.AC↗

arXiv2022

Cofiniteness with respect to extension of Serre subcategories

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\mathcal{S}$ be an arbitrary Serre subcategory of $R$-modules satisfying the condition $C_{\frak a}$ and let $\mathcal{N}$ be the subcategory of finitely generated $R$-modules. In this paper, we define and study $\mathcal{NS}$-$\frak a$-cofinite modules with respect to the extension subcategory $\mathcal{NS}$ as an generalization of the classical notion, namely $\frak a$-cofinite modules. For the lower dimensions, we show that the classical results of $\frak a$-cofiniteness hold for the new notion.

math.AC↗

arXiv2022

Cofiniteness with respect to extension of Serre subcategories at small dimensions

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\cS$ be an arbitrary Serre subcategory of $R$-modules and let $\cN$ be the subcategory of finitely generated $R$-modules. In this paper, we study $\cN\cS$-$\frak a$-cofinite modules with respect to the extension subcategory $\cN\cS$ when $\dim R/\frak a\leq 2$. We also study $\frak a$-cofiniteness with respect to a new dimension.

math.AC↗

arXiv2022

A criterion for cofiniteness of modules

Let $A$ be a commutative noetherian ring, $\frak a$ be an ideal of $A$, $m,n$ be non-negative integers and let $M$ be an $A$-module such that $\Ext^i_A(A/\frak a,M)$ is finitely generated for all $i\leq m+n$. We define a class $\cS_n(\frak a)$ of modules and we assume that $H_{\frak a}^s(M)\in\cS_{n}(\frak a)$ for all $s\leq m$. We show that $H_{\frak a}^s(M)$ is $\frak a$-cofinite for all $s\leq m$ if either $n=1$ or $n\geq 2$ and $\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$ is finitely generated for all $1\leq t\leq n-1$, $i\leq t-1$ and $s\leq m$. If $A$ is a ring of dimension $d$ and $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of dimension $\leq d-1$, then we prove that $M\in\cS_n(\frak a)$ for any ideal $\frak a$ of $A$.

math.AC↗

arXiv2022

Melkersson condition for extension of Serre subcategories

Let $R$ be a commutative noetherian ring and let $\frak a$ be an ideal of $R$. In this paper, we study a certain condition, namely $C_{\frak a}$, introduced by Aghapournahr and Melkersson, on the extension of two subcategories of $R$-modules. We extend and generalize some of the main results of Yoshizawa [Y1,Y2]. As an example of extension of subcategories, we study the weakly Laskerian modules and we find some conditions under which the local cohomology modules of a weakly Laskerian module lie in an arbitrary Serre subcategory. Eventually, we investigate the cofiniteness of the local cohomology modules of weakly Laskerian modules.

math.AC↗

arXiv2020

Torsion theory of coherent functors

Let $C$ be an additive category with cokernels and let Mod($C$) be the category of additive functors from $C^{op}$ to the category Ab of abelian groups. Let mod($C$) be the full subcategory of Mod($C$) consisting of coherent functors. In this paper, we first study some basic properties of pseudo-kernel of morphisms in $C$. When $C$ has pseudo-kernels, mod($C$) is abelian and then, in this case, we study radical functors, half exact functors, left exact functors and injective objects in mod($C$). At last, we extend the results for Mod($C$).

math.CT↗

arXiv2020

A new dimension for Grothendieck categories via the atom spectrum

In this paper, we define a new dimension for objects in a Grothendieck category $\mathcal{A}$. We show that it serves as a lower bound for Gabriel-Krull dimension and under certain conditions, the two dimensions coincide. We carry out our investigation for a fully right bounded ring $A$. We introduce a new spectrum Comp$\,A$ via compressible right $A$-modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right $A$-modules can be computed via the length of chain of prime ideals of $A$ and also the length of chain of elements of Comp$\,A$.

math.CT↗

arXiv2019

Local cohomology in Grothendieck categories

Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper we define and study the section functor on $\mathcal{A}$ with respect to an open subset of ASpec$\mathcal{A}$. Next we define and study local cohomology theory in $\mathcal{A}$ in terms of the section functors. Finally we study abstract local cohomology functor on the derived category $\mathcal{D}^+(\mathcal{A})$.

math.AC↗

arXiv2017

Some finiteness properties of generalized graded local cohomology modules

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 Artinianess of $Γ_{\mathfrak m_0R}(H_{R_+}^i(M,N)), H_{\mathfrak m_0R}^1(H_{R_+}^i(M,N)), H_{R_+}^i(M,N)/{\mathfrak m_0}H_{R_+}^i(M,N), H_{R_+}^j(M,H_{\mathfrak m_0R}^i(N)), H_{\mathfrak m_0R}^j(M,H_{R_+}^i(N))$, where $R_+$ denotes the irrelevant ideal of $R$.

math.AC↗

arXiv2017

Some conditions under which certain types of modules possess localization property

By a localization property we mean a property that preserved under localizing modules at multiplicative closed sets. The aim of this paper is to find conditions under which we can transfere a certain property of a given module to its localization at multiplicative closed sets and conversely, that means we determine those conditions which make a given property of a module as a localization property, so we give several conditions under which certain types of modules posses localization property.

math.AC↗

arXiv2014

Melkersson condition on Serre subcategories

Let $R$ be a commutative noetherian ring, let $\frak a$ and $\frak b$ be two ideals of $R$; and let $\Ss$ be a Serre subcategory of $R$-modules. We give a necessary and sufficient condition by which $\Ss$ satisfies $C_{\frak a}$ and $C_{\frak b}$ conditions. As an conclusion we show that over a artinian local ring, every Serre subcategory satisfies $C_{\frak a}$ condition. We also show that $\Ss_{\frak a}$ is closed under extension of modules. If $\Ss$ is a torsion subcategory, we prove that $S$ satisfies $C_{\frak a}$ condition. We prove that $C_{\frak a}$ condition can be transferred via rings homomorphism. As some applications, we give several results concerning with Serre subcategories in local cohomology theory.

math.AC↗

arXiv2011

Gorenstein injective dimension and a generalization of Ischebeck Formula

Let $(R,\frak m)$ be a commutative Noetherian local ring and let $M$ and $N$ be finitely generated $R$-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper we prove a generalization of Ischebeck Formula, that is $\depth_RM+\sup\{i| {0.1cm}\Ext_R^i(M,N)\neq 0\}=\depth R.$

math.AC↗