Searcharxiv⌕ Search

arXiv subjects

Kamran Divaani-Aazar

Publications and source records attributed to Kamran Divaani-Aazar.

At least 19 recordsLinked to original sources
arXiv2026

Regularity, local cohomology and injective modules under contracting endomorphisms

This work extends several fundamental results previously established for rings of characteristic $p$ to the broader class of rings that admit a contracting endomorphism. Specifically, let $ϕ: R \to R$ be such an endomorphism, and denote by $^ϕR$ the ring $R$ endowed with the left $R$-module structure induced by $ϕ$. We employ the functor $\text{Hom}_R(^ϕR,-)$ to derive new characterizations of regular rings. Furthermore, we examine the behavior of the functor $(-) \otimes_R {^ϕ}R$ on local cohomology modules and injective modules.

math.AC↗
arXiv2026

Cotilting invariance of the Auslander-Reiten conjecture

Let R be an associative ring with identity, and let T be a tilting right R-module, with S=End(T). It is known that if R is a Noetherian algebra that satisfies the Auslander-Reiten conjecture, then so is S. In this paper, we consider the dual situation where C is a cotilting right R-module, with S=End(C). We investigate the invariance of the property of satisfying the Auslander-Reiten conjecture when passing from R to S.

math.RT↗
arXiv2025

Comparing four definitions of cotilting modules

In contrast to the theory of tilting modules, the dual theory lacks a unified definition. Nevertheless, several notions of cotilting modules have been proposed. In this paper, we compare four of the main definitions of cotilting modules that have appeared in the literature. We show that, in the setting of finitely generated right modules, three of these definitions coincide over right Artinian Noetherian algebras, and all four coincide over Artin algebras.

math.RT↗
arXiv2025

On the Wakamatsu tilting conjecture

Let R be an associative ring with identity. We establish that the generalized Auslander-Reiten conjecture implies the Wakamatsu tilting conjecture. Furthermore, we prove that any Wakamatsu tilting R-module of finite projective dimension that is tensorly faithful is projective. By utilizing this result, we show the validity of the Wakamatsu tilting conjecture for R in two cases: when R is a left Artinian local ring or when it is the group ring of a finite group G over a commutative Artinian ring.

math.RT↗
arXiv2025

Cotilting modules and Gorenstein homological dimensions

For a dualizing module $D$ over a commutative Noetherian ring $R$ with identity, it is known that its Auslander class $\mathscr{A}_D\left(R\right)$ (respectively, Bass class $\mathscr{B}_D\left(R\right)$) is characterized as those $R$-modules with finite Gorenstein flat dimension (respectively, finite Gorenstein injective dimension). We establish an analogue of this result in the context of cotilting modules over general Noetherain rings.

math.RT↗
arXiv2024

Characterization and examples of commutative isoartinian rings

Noetherian rings have played a fundamental role in commutative algebra, algebraic number theory, and algebraic geometry. Along with their dual, Artinian rings, they have many generalizations, including the notions of isonoetherian and isoartinian rings. In this paper, we prove that the Krull dimension of every isoartinian ring is at most one. We then use this result to provide a characterization of isoartinian rings. Specifically, we prove that a ring $R$ is isoartinian if and only if $R$ is uniquely isomorphic to the direct product of a finite number of rings of the following types: (i) Artinian local rings; (ii) non-Noetherian isoartinian local rings with a nilpotent maximal ideal; (iii) non-field principal ideal domains; (iv) Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (v) non-Noetherian isoartinian rings $A$ with $\Min A$ being a singleton and $\Min A \subsetneq \Ass A$; (vi) non-Noetherian isoartinian rings $A$ with a unique element in $\Min A$ that is not maximal, and $\Min A=\Ass A$. Several examples of these types of rings are also provided.

math.AC↗
arXiv2023

Dualities and equivalences of the category of relative Cohen-Macaulay modules

In this paper, we establish the global analogues of some dualities and equivalences in local algebra by developing the theory of relative Cohen-Macaulay modules. Let R be a commutative Noetherian ring (not necessarily local) with identity and a a proper ideal of R. The notions of a-relative dualizing modules and a-relative big Cohen-Macaulay modules are introduced. With the help of a-relative dualizing modules, we establish the global analogue of the duality on the subcategory of Cohen-Macaulay modules in local algebra. Lastly, we investigate the behavior of the subcategory of a-relative Cohen-Macaulay modules and a-relative generalized Cohen-Macaulay modules under Foxby equivalence.

math.AC↗
arXiv2023

Relative homological rings and modules

The study of rings and modules with homological criteria is a cornerstone of commutative algebra. Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. In this paper, a relative analogue of the theory of homological rings and modules is developed. We introduce the notions of $\frak a$-relative regular, $\frak a$-relative complete intersection, and $\frak a$-relative Gorenstein rings and modules. We extend some classical results by demonstrating some interactions between these types of rings and modules.

math.AC↗
arXiv2022

Relative Cohen-Macaulay modules under ring homomorphisms

Let $R$ be a commutative Noetherian ring with identity (not necessarily local) and $\frak a$ a proper ideal of $R$. We study the invariance of some classes of $\frak a$-relative Cohen-Macaulay modules under pure ring homomorphisms and ring homomorphisms of finite flat dimension. Our results extend several results in the existing literature on homological modules.

math.AC↗
arXiv2022

The balance of relative Ext groups defined by semi dualizing modules

Let $R$ be a commutative Noetherian ring with identity and $C$ a semidualizing module for $R$. Let $\mathscr{P}_C(R)$ and $\mathscr{I}_C (R)$ denote, respectively, the classes of $C$-projective and $C$-injective $R$-modules. We show that their induced Ext bifunctors $\text{Ext}^i_{\mathscr{P}_C}(-,\sim)$ and $\text{Ext}^i_{\mathscr{I}_C}(-,\sim)$ coincide for all $i\geq 0$ if and only if $C$ is projective. Also, we provide some other criteria for $C$ to be projective by using some special cotorsion theories.

math.AC↗
arXiv2022

Characterizing certain semidualizing complexes via their Betti and Bass numbers

It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=\sup C$ and $d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if and only if the $n$th Betti number of $C$ is one. Also, we show that $C$ is a dualizing complex for $R$ if and only if the $d$th Bass number of $C$ is one.

math.AC↗
arXiv2021

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In particular, we show that generalized Cohen-Macaulay R-modules are invariant under this equivalence and if M is a finitely generated R-module in the Auslander class with respect to C such that C\otimes_RM is surjective Buchsbaum, then M is also surjective Buchsbaum.

math.AC↗
arXiv2021

Modules whose finiteness dimensions coincide with their cohomological dimensions

Let a be an ideal of a commutative Noetherian ring R with identity. We study finitely generated R-modules M whose a-finiteness and a-cohomological dimensions are equal. In particular, we examine relative analogues of quasi-Buchsbaum, Buchsbaum and surjective Buchsbaum modules. We reveal several interactions between these types of modules that extend some of the existing results in the classical theory to the relative one.

math.AC↗
arXiv2019

On the New Intersection Theorem for totally reflexive modules

Let (R,m,k) be a local ring. We establish a totally reflexive analogue of the New Intersection Theorem, provided for every totally reflexive R-module M, there is a big Cohen-Macaulay R-module B_M such that the socle of B_M\otimes_RM is zero. When R is a quasi-specialization of a G-regular local ring or when M has complete intersection dimension zero, we show the existence of such a big Cohen-Macaulay R-module. It is conjectured that if R admits a non-zero Cohen-Macaulay module of finite Gorenstein dimension, then it is Cohen-Macaulay. We prove this conjecture if either R is a quasi-specialization of a G-regular local ring or a quasi-Buchsbaum local ring.

math.AC↗
arXiv2019

Cohomological dimension and relative Cohen-Maculayness

Let R be a commutative Noetherian (not necessarily local) ring with identity and a be a proper ideal of R. We introduce a notion of a-relative system of parameters and characterize them by using the notion of cohomological dimension. Also, we present a criterion of relative Cohen-Macaulay modules via relative system of parameters.

math.AC↗
arXiv2018

A New Outlook on Cofiniteness

Let $\mathfrak{a}$ be an ideal of a commutative noetherian (not necessarily local) ring $R$. In the case $\cd(\mathfrak{a},R)\leq 1$, we show that the subcategory of $\mathfrak{a}$-cofinite $R$-modules is abelian. Using this and the technique of way-out functors, we show that if $\cd(\mathfrak{a},R)\leq 1$, or $\dim(R/\mathfrak{a}) \leq 1$, or $\dim(R) \leq 2$, then the local cohomology module $H^{i}_{\mathfrak{a}}(X)$ is $\mathfrak{a}$-cofinite for every $R$-complex $X$ with finitely generated homology modules and every $i \in \mathbb{Z}$. We further answer Question 1.3 in the three aforementioned cases, and reveal a correlation between Questions 1.1, 1.2, and 1.3.

math.AC↗