arXiv subjects

Maryam Jahangiri

Publications and source records attributed to Maryam Jahangiri.

14 recordsLinked to original sources

arXiv2026

Koszul cohomology of Čech cohomology modules

Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\ldots,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\ldots,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf y)$. Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf x$ and the Čech complex with respect to $\mathbf y$. Using the two spectral sequences associated with this double complex, we prove that \[ Ext_R^i(R/(\mathbf x),H_I^j(R))\in\mathcal S \quad\text{for all }i,j\in \mathbb N_0 \] implies \[ H_I^j(R/(\mathbf x))\in\mathcal S \quad\text{for all }j\in\mathbb N_0. \]

math.AC↗

arXiv2026

Bass numbers of graded components of local cohomology modules

Let $R=\bigoplus_{n\in \NN_0}R_n$ be a standard graded ring, $R_+=\bigoplus_{n\in \NN}R_n$ its irrelevant ideal, and $M$ a finitely generated graded $R$-module. In this paper, we study the asymptotic behavior of the sequence $\{μ^i(\p_0, H^j_{R_+}(M)_n)\}_{n\in \Z}$ of Bass numbers of graded components of local cohomology modules with respect to an ideal $\p_0\in \Spec(R_0)$ in each of the following cases: (1) $i=0$ or $i= 1$ and $j\leq f_{R_+}(M)$, (2) $R_0$ is regular, $i= \hei(\p_0)$ or $i= \hei(\p_0)- 1$ and $j= \cd_{R_+}(M)$, (3) $M$ is relative Cohen-Macaulay with respect to $R_+$. Here, $\cd_{R_+}(M)$ and $f_{R_+}(M)$ denote the cohomological dimension and finiteness dimension of $M$ with respect to $R_+$, respectively.

math.AC↗

arXiv2021

Asymptotic behaviour of graded local cohomology modules via linkage

Assume that $R=\oplus_{n\in \mathbb{N}_0}R_n$ is a standard graded algebra over the local ring $(R_0,\mathfrak{m}_0)$, $\mathfrak{a}$ is a homogeneous ideal of $R$, $M$ is a finitely generated graded $R$-module and $R_+:=\oplus_{n\in \mathbb{N}}R_n$ denotes the irrelevant ideal of $R$. In this paper, we study the asymptotic behaviour of the set $\{ \operatorname{grade}(\mathfrak{a} \cap R_0, H^{\operatorname{grade}(R_+,M)}_{R_+}(M)_n) \}_{n \in \mathbb{Z}}$ as $n \rightarrow -\infty$, in the case where $\mathfrak{a}$ and $R_+$ are homogenously linked over $M$.

math.AC↗

arXiv2021

Graded local cohomology modules with respect to the linked ideals

Let $R=\oplus_{n\in \N_0}R_n$ be a standard graded ring, $M$ be a finitely generated graded $R$-module and $R_+:=\oplus_{n\in \N}R_n$ denotes the irrelevant ideal of $R$. In this paper, considering the new concept of linkage of ideals over a module, we study the graded components $H^i_{\fa}(M)_n$ when $\fa$ is an h-linked ideal over $M$. More precisely, we show that $H^i_{\fa}(M)$ is tame in each of the following cases: \begin{itemize} \item [(i)] $i=f_{\fa}^{R_+}(M)$, the first integer $i$ for which $R_+\nsubseteq \sqrt{0:H^i_{\fa}(M)}$; \item [(ii)] $i=\cd(R_+,M)$, the last integer $i$ for which $H^{i}_{R_+}(M)\neq 0$, and $\fa=\fb+R_+$ where $\fb$ is an h-linked ideal with $R_+$ over $M$. \end{itemize} Also, among other things, we describe the components $H^i_{\fa}(M)_n$ where $\fa$ is radically h-$M$-licci with respect to $R_+$ of length 2.

math.AC↗

arXiv2019

Cohomological dimension with respect to the linked ideals

Let $R$ be a commutative Noetherian ring. Using the new concept of linkage of ideals over a module, we show that if $\mathfrak{a}$ is an ideal of $R$ which is linked by the ideal $I$, then $cd(\mathfrak{a},R) \in \{ grad \mathfrak{a}, cd(\fa, H^{grad \mathfrak{a}}_ {\mathfrak{c}} (R)) + grad \mathfrak{a}\}, $ where $\mathfrak{c} : = \bigcap_{\mathfrak{p} \in Ass \frac{R}{I}- V(\mathfrak{a})}\mathfrak{p}$. Also, it is shown that for every ideal $\mathfrak{b}$ which is geometrically linked with $\mathfrak{a},$ $cd(\mathfrak{a}, H^{grad \mathfrak{b}}_ {\mathfrak{b}} (R))$ does not depend on $\mathfrak{b}$

math.AC↗

arXiv2018

Linkage of ideals over a module

Inspired by the works in linkage theory of ideals, we define the concept of linkage of ideals over a module. Several known theorems in linkage theory are improved or recovered by new approaches. Specially, we make some extensions and generalizations of the basic result of Peskine and Szpiro \cite[prop 1.3]{PS}, namely if $R$ is a Gorenstain local ring, $\mathfrak{a} \neq 0$ (an ideal of $R$) and $\mathfrak{b} := 0:_R \mathfrak{a}$ then $\frac{R}{\mathfrak{a}}$ is Cohen-Macaulay if and only if $\frac{R}{\mathfrak{a}}$ is unmixed and $\frac{R}{\mathfrak{b}}$ is Cohen-Macaulay.

math.AC↗

arXiv2018

Attached and Assoiciated Primes Of Local Cohomology Modules Via Linkage

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules of $M$ with respect to some linked ideals over it.

math.AC↗

arXiv2018

Characterization of some special rings via linkage

Some descriptions of linked ideals in a commutative Notherian ring $R$ are provided in terms of the Associated prime ideals of $R$. Then, among other things, we make some characterization of Cohen-Macaulay, Gorenstein and regular local rings in terms of their linked ideals.

math.AC↗

arXiv2017

On the rate of graded modules

Let $K$ be a field, $R$ a standard graded $K$-algebra and $M$ be a finitely generated graded $R$-module. The rate of $M$, $rate_R(M)$, is a measure of the growth of the shifts in the minimal graded free resolution of $M$. In this paper, we find upper bounds for this invariant. More precisely, let $(A,\mathfrak{n})$ be a regular local ring and $I\subseteq \mathfrak{n} ^t$ be an ideal of $A$, where $t\geq 2$. We prove that if $(B=A/I, \mathfrak{m} =\mathfrak{n} /I)$ is a Cohen-Macaulay local ring with multiplicity $e(B)= \binom{h+t-1}{h}$, where $h=embdim(B)-dim B$, then $rat(gr_{\mathfrak{m}}(B))=t-1$ and for every $B$-module $N$, which annihilated by a minimal reduction of $\mathfrak{m}$, $rate_{gr_{\mathfrak{m}}(B)}(gr_{\mathfrak{m}}(N))\leq t-1$.

math.AC↗

arXiv2013

Castelnuovo-Mumford regularity and cohomological dimension

Let $R=\oplus_{i\in \N_0}R_n$ be a standard graded ring, $R_+ :=\oplus_{i\in \N}R_n$ be the irrelevant ideal of $R$ and $\fa_0$ be an ideal of $R_0$. In this paper, as a generalization of the concept of Castelnouvo-Mumford regularity $\reg(M)$ of a finitely generated graded $R$-module $M$, we define the regularity of $M$ with respect to $\fa_0+ R_+$, say $\reg_{\fa_0+ R_+}(M)$. And we study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated $R_0$-modules. Also, we will express $\reg_{\fa_0+ R_+}(M)$ in terms of some invariants of the minimal graded free resolution of $M$ and see that in a special case this invariant is independent of the choice of $\fa_0$.

math.AC↗

arXiv2011

Tame Loci of Certain Local Cohomology Modules

Let $M$ be a finitely generated graded module over a Noetherian homogeneous ring $R = \bigoplus_{n \in \mathbb{N}_0}R_n$. For each $i \in \mathbb{N}_0$ let $H^i_{R_{+}}(M)$ denote the $i$-th local cohomology module of $M$ with respect to the irrelevant ideal $R_+ = \bigoplus_{n > 0} R_n$ of $R$, furnished with its natural grading. We study the tame loci $\ft^i(M)^{\leq 3}$ at level $i \in \mathbb{N}_0$ in codimension $\leq 3$ of $M$, that is the sets of all primes $\fp_0 \subset R_0$ of height $\leq 3$ such that the graded $R_{\fp_0}$-modules $H^i_{R_{+}}(M)_{\fp_0}$ are tame.

math.AC↗

arXiv2011

Relative Cohen-Macaulayness and relative unmixedness of bigraded modules

In this paper we study the finitely generated bigraded modules over a standard bigraded polynomial ring which are relative Cohen-Macaulay or relative unmixed with respect to one of the irrelevant bigraded ideals. A generalization of Reisner's criterion for Cohen-Macaulay simplicial complexes is considered.

math.AC↗

arXiv2009

Boundedness of Cohomology

Let $d \in \N$ and let $\D^d$ denote the class of all pairs $(R,M)$ in which $R = \bigoplus_{n \in \N_0} R_n$ is a Noetherian homogeneous ring with Artinian base ring $R_0$ and such that $M$ is a finitely generated graded $R$-module of dimension $\leq d$. The cohomology table of a pair $(R,M) \in \D^d$ is defined as the family of non-negative integers $d_M:= (d^i_M(n))_{(i,n) \in \N \times \Z}$. We say that a subclass $\mathcal{C}$ of $\D^d$ is of finite cohomology if the set $\{d_M \mid (R,M) \in \C\}$ is finite. A set $\mathbb{S} \subseteq \{0,... ,d-1\}\times \Z$ is said to bound cohomology, if for each family $(h^σ)_{σ\in \mathbb{S}}$ of non-negative integers, the class $\{(R,M) \in \D^d\mid d^i_M(n) \leq h^{(i,n)} {for all} (i,n) \in \mathbb{S}\}$ is of finite cohomology. Our main result says that this is the case if and only if $\mathbb{S}$ contains a quasi diagonal, that is a set of the form $\{(i,n_i)| i=0,..., d-1\}$ with integers $n_0> n_1 > ... > n_{d-1}$. We draw a number of conclusions of this boundedness criterion.

math.AC↗

arXiv2009

Castelnuovo-Mumford regularity of deficiency modules

Let $d \in \N$ and let $M$ be a finitely generated graded module of dimension $\leq d$ over a Noetherian homogeneous ring $R$ with local Artinian base ring $R_0$. Let $\beg(M)$, $\gendeg(M)$ and $\reg(M)$ respectively denote the beginning, the generating degree and the Castelnuovo-Mumford regularity of $M$. If $i \in \N_0$ and $n \in Z$, let $d^i_M(n)$ denote the $R_0$-length of the $n$-th graded component of the $i$-th $R_+$-transform module $D^i_{R_+}(M)$ of $M$ and let $K^i(M)$ denote the $i$-th deficiency module of $M$. Our main result says, that $\reg(K^i(M))$ is bounded in terms of $\beg(M)$ and the "diagonal values" $d^j_M(-j)$ with $j = 0,..., d-1$. As an application of this we get a number of further bounding results for $\reg(K^i(M))$.

math.AC↗