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Mohsen Gheibi

Publications and source records attributed to Mohsen Gheibi.

13 recordsLinked to original sources
arXiv2025

Quasi-homological dimensions with respect to semidualizing modules

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension of a module over commutative Noetherian rings, a homological invariant extending the classic projective dimension of a module, and Gheibi later developed the dual notion of quasi-injective dimension. Takahashi and White in 2010 introduced the projective and injective dimension of a module with respect to a semidualizing module, which likewise generalize their classic counterparts. In this paper we unify and extend these theories by defining and studying the quasi-projective and quasi-injective dimension of a module with respect to a semidualizing module. We establish several results generalizing classic formulae such as the Auslander-Buchsbaum formula, Bass' formula, Ischebeck's formula, Auslander's depth formula and Jorgensen's dependency formula. Furthermore, we prove a special case of the Auslander-Reiten conjecture and investigate rigidity properties of Ext and Tor.

math.AC↗
arXiv2024

Characteristic modules over a local ring

Let $R$ be a commutative noetherian local ring, and let $M$ be a finitely generated $R$-module. Inspired by works of Vasconcelos and Briggs on characterization of complete intersection local rings through the homological properties of the conormal module, in this paper, we define the characteristic module T$_M$ and the cocharacteristic module E$_M$ of $M$, and investigate their properties. Our main results include characterizations of Cohen--Macaulay and Gorenstein local rings. Also, we show that if the injective dimension of the conormal module over an almost complete intersection ring is finite, then $R$ is a complete intersection.

math.AC↗
arXiv2023

Quasi-injective dimension

Following our previous work about quasi-projective dimension, in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module $M$ over a local ring $R$ is finite, then it is equal to the depth of $R$. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then $R$ is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then $R$ is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.

math.AC↗
arXiv2023

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↗
arXiv2022

The homotopy Lie algebra of a Tor-independent tensor product

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $π(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $π(S)$ is the pullback of (adjusted) Lie algebras along the maps $π(S_i)\to π(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of $R,S_1,S_2$ and $S$.

math.AC↗
arXiv2021

Quasi-projective dimension

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is a generalization of projective dimension. We discuss various properties of quasi-projective dimension. Among other things, we prove the following. (1) Over a quotient of a regular local ring by a regular sequence, every finitely generated module has finite quasi-projective dimension. (2) The Auslander--Buchsbaum formula and the depth formula for modules of finite projective dimension remain valid for modules of finite quasi-projective dimension. (3) Several results on vanishing of Tor and Ext hold for modules of finite quasi-projective dimension.

math.AC↗
arXiv2018

Totally Reflexive Modules and Poincaré Series

We study Cohen-Macaulay non-Gorenstein local rings $(R,\mathfrak{m},k)$ admitting certain totally reflexive modules. More precisely, we give a description of the Poincaré series of $k$ by using the Poincaré series of a non-zero totally reflexive module with minimal multiplicity. Our results generalize a result of Yoshino to higher-dimensional Cohen-Macaulay local rings. Moreover, from a quasi-Gorenstein ideal satisfying some conditions, we construct a family of non-isomorphic indecomposable totally reflexive modules having an arbitrarily large minimal number of generators.

math.AC↗
arXiv2017

Associated Primes and Syzygies of Linked Modules

Motivated by the notion of geometrically linked ideals, we show that over a Gorenstein local ring $R$, if a Cohen-Macaulay $R$-module $M$ of grade $g$ is linked to an $R$-module $N$ by a Gorenstein ideal $c$, such that $Ass_R(M)\cap Ass_R(N)=\emptyset$, then $M\otimes_RN$ is isomorphic to direct sum of copies of $R/a$, where $a$ is a Gorenstein ideal of $R$ of grade $g+1$. We give a criterion for the depth of a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of the residue field $k$. As a result we characterize a local ring $(R,m,k)$ in terms of the homological dimensions of the modules linked to the syzygies of $k$.

math.AC↗
arXiv2015

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↗
arXiv2013

Brauer-Thrall for totally reflexive modules over local rings of higher dimension

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive $R$-modules of multiplicity $n$.

math.AC↗
arXiv2012

Sequence of Exact Zero-Divisors

The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studied. Some properties of sequences of exact zero-divisors are compared with regular sequences.

math.AC↗
arXiv2010

Linkage of modules over Cohen-Macaulay rings

Inspired by the works in linkage theory of ideals, the concept of sliding depth of extension modules is defined to prove the Cohen-Macaulyness of linked module if the base ring is merely Cohen-Macaulay. Some relations between this new condition and other module-theory conditions such as G-dimension and sequentially Cohen-Macaulay are established. By the way several already known theorems in linkage theory are improved or recovered by new approaches.

math.AC↗