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Somayeh Moradi

Publications and source records attributed to Somayeh Moradi.

At least 37 records · Page 2Linked to original sources
arXiv2023

Ideals with componentwise linear powers

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of $A$ are componentwise linear. Applying this result to the Rees ring $\mathcal{R}(I)$ of a graded ideal $I$ gives a criterion on $I$ to have componentwise linear powers. Moreover, for any given graph $G$, a construction on $G$ is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This in particular implies that for any Cohen-Macaulay Cameron-Walker graph $G$ all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs and Cohen-Macaulay bipartite graphs produces cover ideals with componentwise linear powers.

math.AC↗
arXiv2022

Toric rings arising from vertex cover ideals

We extend the sortability concept to monomial ideals which are not necessarily generated in one degree and as an application we obtain normal Cohen-Macaulay toric rings attached to vertex cover ideals of graphs. Moreover, we consider a construction on a graph called a clique multi-whiskering which always produces vertex cover ideals with componentwise linear powers.

math.AC↗
arXiv2021

Some homological properties of Borel type ideals

We study ideals of Borel type, including $k$-Borel ideals and $t$-spread Veronese ideals. We determine their free resolutions and their homological shift ideals. The multiplicity and the analytic spread of equigenerated squarefree principal Borel ideals are computed. For the multiplicity, the result is given under an additional assumption which is always satisfied for squarefree principal Borel ideals. These results are used to analyze the behaviour of height, multiplicity and analytic spread of the homological shift ideals $HS_j(I)$ as functions of $j$, when $I$ is an equigenerated squarefree Borel ideal.

math.AC↗
arXiv2021

Rings of Teter type

Let $R$ be a Cohen--Macaulay local $K$-algebra or a standard graded $K$-algebra over a field $K$ with a canonical module $ω_R$. The trace of $ω_R$ is the ideal $tr(ω_R)$ of $R$ which is the sum of those ideals $φ(ω_R)$ with $φ\in Hom_R(ω_R,R)$. The smallest number $s$ for which there exist $φ_1, \ldots, φ_s \in Hom_R(ω_R,R)$ with $tr(ω_R)=φ_1(ω_R) + \cdots + φ_s(ω_R)$ is called the Teter number of $R$. We say that $R$ is of Teter type if $s = 1$. It is shown that $R$ is not of Teter type if $R$ is generically Gorenstein. In the present paper, we focus especially on $0$-dimensional graded and monomial $K$-algebras and present various classes of such algebras which are of Teter type.

math.AC↗
arXiv2021

Graded ideals of König type

Inspired by the notion of König graphs we introduce graded ideals of König type with respect to a monomial order $<$. It is shown that if $I$ is of König type, then the Cohen--Macaulay property of $\ini_<(I)$ does not depend on the characteristic of the base field. This happens to be the case also for $I$ itself when $I$ is a binomial edge ideal. Attached to an ideal of König type is a sequence of linear forms, whose elements are variables or differences of variables. This sequence is a system of parameters for $\ini_<(I)$, and is a potential system of parameters for $I$ itself. We study in detail the ideals of König type among the edge ideals, binomial edge ideals and the toric ideal of a Hibi ring and use the König property to determine explicitly their canonical module.

math.AC↗
arXiv2020

Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $R(I)$ of a graded ideal or the symmetric algebra $Sym(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.

math.AC↗
arXiv2020

Systems of parameters and the Cohen--Macaulay property

We recall a numerical criteria for Cohen--Macaulayness related to system of parameters, and introduce monomial ideals of König type which include the edge ideals of König graphs. We show that a monomial ideal is of König type if and only if its corresponding residue class ring admits a system of parameters whose elements are of the form $x_i-x_j$. This provides an algebraic characterization of König graphs. We use this special parameter systems for the study of the edge ideal of König graphs and the study of the order complex of a certain family of posets. Finally, for any simplicial complex $Δ$ we introduce a system of parameters for $K[Δ]$ with a universal construction principle, independent of the base field and only dependent on the faces of $Δ$. This system of parameters is an efficient tool to test Cohen--Macaulayness of the Stanley--Reisner ring of a simplicial complex.

math.AC↗
arXiv2020

Closed neighborhood ideal of a graph

We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhood of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal of the graph. We study some algebraic invariants of these ideals like Castelnuovo-Mumford regularity and projective dimension and present some combinatorial descriptions for these invariants in terms of graph invariants.

math.AC↗
arXiv2020

Homological shift ideals

For a monomial ideal $I$, we consider the $i$th homological shift ideal of $I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th multigraded shifts of $I$. Some algebraic properties of this ideal are studied. It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$, $\text{HS}_i(I(P))\subseteq \text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the monomial localization of $I$. In particular, we consider the homological shift ideal of some families of monomial ideals with linear quotients. For any $\textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of complement of any path graph, it is proved that $\text{HS}_i(I)$ has linear quotients for all $i$. As an example of $\textbf{c}$-bounded principal Borel ideals, Veronese type ideals are considered and it is shown that the homological shift ideal of these ideals are polymatroidal. This implies that for any polymatroidal ideal which satisfies the strong exchange property, $\text{HS}_j(I)$ is again a polymatroidal ideal for all $j$. Moreover, for any edge ideal with linear resolution, the ideal $\text{HS}_j(I)$ is characterized and it is shown that $\text{HS}_1(I)$ has linear quotients.

math.AC↗
arXiv2020

Some interpretations for algebraic invariants of edge ideals of hypergraphs via combinatorial invariants

The present work is concerned with characterizing some algebraic invariants of edge ideals of hypergraphs. To this aim, firstly, we introduce some kinds of combinatorial invariants similar to matching numbers for hypergraphs. Then we compare them to each other and to previously existing ones. These invariants are used for characterizing or bounding some algebraic invariants of edge ideals of hypergraphs such as graded Betti numbers, projective dimension and Castelnouvo-Mumford regularity.

math.AC↗
arXiv2019

Sortable simplicial complexes and $t$-independence ideals of proper interval graphs

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the graphs whose independence complex is sortable. By using this characterization, we show that the ideal generated by all squarefree monomials corresponding to independent sets of vertices of $G$ of size $t$ (for a given positive integer $t$) has the strong persistence property, when $G$ is a proper interval graph. Moreover, all of its powers have linear quotients.

math.AC↗
arXiv2019

On the monomial reduction number of a monomial ideal in $K[x,y]$

The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.

math.AC↗
arXiv2018

Total mixed domination in graphs

For a graph $G=(V,E)$, we call a subset $ S\subseteq V \cup E$ a total mixed dominating set of $G$ if each element of $V \cup E$ is either adjacent or incident to an element of $S$, and the total mixed domination number $γ_{tm}(G)$ of $G$ is the minimum cardinality of a total mixed dominating set of $G$. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.

math.CO↗
arXiv2018

A family of monomial ideals with the persistence property

In this paper we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I=Ind_t(P_n)$ generated by all monomials $\textbf{x} ^F$, where $F$ is an independent set of vertices of the path graph $P_n$ of size $t$, which is indeed the facet ideal of the $t$-th skeleton of the independence complex of $P_n$. We describe the set of associated primes of all powers of $I$ explicitly. It turns out that any such ideal $I$ has the persistence property. Moreover the index of stability of $I$ and the stable set of associated prime ideals of $I$ are determined.

math.AC↗
arXiv2017

$t$-clique ideal and $t$-independence ideal of a graph

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover some families of graphs for which the quotient ring of their clique ideal is Cohen-Macaulay are introduced and some homological invariants of the clique ideal of a graph $G$ which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs are studied.

math.AC↗
arXiv2017

On the facet ideal of an expanded simplicial complex

For a simplicial complex $Δ$, the affect of the expansion functor on combinatorial properties of $Δ$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $I(Δ)$ and its Alexander dual which we denote by $J_Δ$ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion $Δ^α$ the ideals $J_Δ$ and $J_{Δ^α}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals $I(Δ)$ and $I(Δ^α)$ are equal. Moreover, the projective dimensions of $I(Δ)$ and $I(Δ^α)$ are compared. In the sequel for a graph $G$, some properties that are equivalent in $G$ and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph $G$, we give some conditions for adding or removing a vertex from $G$, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).

math.AC↗
arXiv2016

On the Stanley-Reisner ideal of an expanded simplicial complex

Let $Δ$ be a simplicial complex. We study the expansions of $Δ$ mainly to see how the algebraic and combinatorial properties of $Δ$ and its expansions are related to each other. It is shown that $Δ$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $Δ$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of $Δ$ and those of their expansions are compared.

math.AC↗