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Majid Eghbali

Publications and source records attributed to Majid Eghbali.

14 recordsLinked to original sources
arXiv2026

Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings

We study the surjectivity of certain maps involving local cohomology modules, which we can realize as a dual version of part of the investigation developed by Bhatt, Blickle, Lyubeznik, Singh and Zhang on the sheaf cohomology of thickenings (i.e., subschemes defined by powers of ideals), where injectivity played a central role. To this end, we introduce and investigate properties of cohomologically Mittag-Leffler (cML) rings, associated to a given flat local endomorphism (for instance the Frobenius map of a regular ring of prime characteristic), a class which we show to contain, in our setting, the so-called cohomologically full rings of Dao, De Stefani and Ma (in particular, Cohen-Macaulay, Stanley-Reisner, and Du Bois singularities) as well as rings with an ideal inducing a pure endomorphism of the quotient. Our two major specific goals rely upon the prime characteristic setting. First, we extend for the class of cML rings a classical result of Peskine and Szpiro that relates the cohomological dimension and the height of a given Cohen-Macaulay ideal. Second, we prove and illustrate a Kodaira type vanishing result on the sheaf cohomology of thickenings.

math.AC↗
arXiv2026

On Hellus--Lyubeznik--Yildirim's conjecture of local cohomology modules

The goal of this paper is to study the so--called Hellus--Lyubeznik--Yildirim (HLY) conjecture, that predicts the following: given a regular local ring $(R,\mathfrak{m})$, and any ideal $I\subset R$, zero is an associated prime ideal of the Matlis dual of any non--zero local cohomology module supported on $I$. Among other results, we give some partial positive answers to this conjecture in the following cases: when $\operatorname{depth} (R/I)=1$, when $\operatorname{depth} (R/I)=2$ under some extra assumptions, when $I$ is a squarefree monomial ideal inside a formal power series ring over a field, and when $R$ is a formal power series over a discrete valuation ring of mixed characteristic.

math.AC↗
arXiv2022

Certain endomorphism rings of local cohomology modules and Lyubeznik numbers

The goal of this paper is twofold; on the one hand, motivated by questions raised by Schenzel, we explore situations where the Hartshorne--Lichtenbaum Vanishing Theorem for local cohomology fails, leading us to simpler expressions of certain local cohomology modules. As application, we give new expressions of the endomorphism ring of these modules. On the other hand, building upon previous work by Àlvarez Montaner, we exhibit the shape of Lyubeznik tables of the so--called partially sequentially Cohen--Macaulay rings as introduced by Sbarra and Strazzanti.

math.AC↗
arXiv2021

On the equality of de Rham depth and formal grade in characteristic zero

Let $Y \subset \mathbb{P}^n_k$ be a non-singular proper closed subset of projective $n$-space over a field $k$ of characteristic zero and let $I \subset R=k[x_0, \ldots, x_n]$ be the homogeneous defining ideal of $Y$. We show that in this case, the de Rham depth of $Y$ is the same as the so-called formal grade of $I$ in $R$.

math.AC↗
arXiv2019

Lyubeznik Tables of Ideals of Cycle Graphs

Let R = K[x_1; : : : ; x_n] be a polynomial ring over a field K, and I := I_C_n be an edge ideal of n-cycle graph C_n. In the present paper, we compute the last column of the Lyubeznik table of R/I.

math.AC↗
arXiv2018

A note on some top local cohomology modules

Let $\fa$ be an ideal of a $d$-dimensional commutative Noetherian ring $R$. In this paper we give some information on some last non-zero local cohomology modules known as top local cohomology modules in particular, $H^{d-1}_{\fa}(R)$.

math.AC↗
arXiv2018

A note on the use of Frobenius map and D-modules in local cohomology

The Frobenius depth denoted by F-depth defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The first aim of the present paper is to compare the F-depth with formal grade and reprove some results of Lyubeznik using formal local cohomology. Then the endomorphism rings of local cohomology modules will be considered. As an application, we reprove the results due to Huneke-Koh in positive characteristic and Lyubeznik in characteristic zero on the annihilators of local cohomology modules.

math.AC↗
arXiv2017

Annihilators of local cohomology modules and simplicity of rings of differential operators

One classical topic in the study of local cohomology is whether the non-vanishing of a specific local cohomology module is equivalent to the vanishing of its annihilator; this has been studied by several authors, including Huneke, Koh, Lyubeznik and Lynch. Motivated by questions raised by Lynch and Zhang, the goal of this paper is to provide some new results about this topic, which provide some partial positive answers to these questions. The main technical tool we exploit is the structure of local cohomology as module over rings of differential operators.

math.AC↗
arXiv2015

On cohomological invariants of local rings in positive characteristic

The Frobenius depth denoted by $\Fdepth$ defined by Hartshorne-Speiser in 1977 and later by Lyubeznik in 2006, in a different way, for rings of positive characteristic. The aim of the present paper is to compare the $\Fdepth$ with formal grade, and $\depth$ to shed more light on the notion of Frobenius depth from a different point of view.

math.AC↗
arXiv2012

On an endomorphism ring of local cohomology

Let $I$ be an ideal of a local ring $(R,\mathfrak m)$ with $d = \dim R.$ For the local cohomology module $H^i_I(R)$ it is a well-known fact that it vanishes for $i > d$ and is an Artinian $R$-module for $i = d.$ In the case that the Hartshorne-Lichtenbaum Vanishing Theorem fails, that is $H^d_I(R) \not= 0,$ we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case $R$ is complete we prove - as a technical tool - that $H^d_I(R) \simeq H^d_{\mathfrak m}(R/J)$ for a certain ideal $J \subset R.$ Thus, properties of $H^d_I(R)$ and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.

math.AC↗