arXiv subjects

Somayeh Moradi

Publications and source records attributed to Somayeh Moradi.

At least 19 recordsLinked to original sources

arXiv2026

Symbolic Rees algebras of complementary edge ideals

Let $G$ be a finite simple graph on $[n]$ and let $I_c(G)$ denote its complementary edge ideal in the polynomial ring $S = K[x_1,\dots,x_n]$. We give a combinatorial description, in terms of the structure of $G$, of the minimal generators of the symbolic Rees algebra $\mathcal{R}_s(I_c(G)) = \bigoplus_{k \geq 0} I_c(G)^{(k)} t^k$, and show that this algebra is generated in degree at most $6$. Moreover, we completely determine the minimal generators of $\mathcal{R}_{s}(I_{c}(G))$ in graph-theoretic terms. We then study in more detail the homological invariants of the symbolic powers $I_c(G)^{(k)}$ for the classes of cycle graphs and complete multipartite graphs. For theses families, we study the behavior of the symbolic depth function $k\mapsto\operatorname{depth} S/I_c(G)^{(k)}$, we obtain the limit depth of the symbolic powers and the Waldschmidt constant of $I_c(G)$, and further prove that all the symbolic powers $I_c(G)^{(k)}$ are componentwise linear.

math.AC↗

arXiv2026

$\textbf{k}$-neighborhood ideals of graphs

In this paper, we introduce and investigate the $\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_1,\ldots,k_n)\in \mathbb{N}^n$ satisfying $1\leq k_i\leq \textrm{deg}_G(i)+1$ for all $i$, the $\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\textrm{NI}_{\textbf{k}}(G)=\sum_{i=1}^n\, (\textbf{x}_W:\, W\subseteq N_G[i],\, |W|=k_i)$$ of $S$, where $\textbf{x}_W=\prod_{i\in W} x_i$. We study homological invariants and properties of $\textrm{NI}_{\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\textrm{NI}_{\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\textrm{NI}_{\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.

math.AC↗

arXiv2026

On homological invariants and Cohen-Macaulayness of closed neighborhood ideals

Let $G$ be a finite simple graph and $NI(G)$ be the closed neighborhood ideal of $G$ in the polynomial ring $S=K[V(G)]$. In this paper, we study the Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness of this ideal. For any chordal graph $G$, we show that $\text{reg}(S/NI(G))=τ(G)$, where $τ(G)$ denotes the vertex cover number of $G$. This generalizes the corresponding result for trees shown in [3], as in trees $τ(G)$ is the same as the matching number of $G$. When $G$ is a bipartite graph or a very well-covered graph, we notice that $\text{reg}(S/NI(G))\geq τ(G)$ and that this inequality can be strict in general. Moreover, we describe the projective dimension of $S/NI(G)$ for some families of graphs. Finally, we give a characterization of very well-covered graphs $G$ for which the ring $S/NI(G)$ is Cohen-Macaulay.

math.AC↗

arXiv2025

Rees algebras of complementary edge ideals

In this paper we investigate the Rees algebras of squarefree monomial ideals $I \subset S=K[x_1,\dots,x_n]$ generated in degree $n-2$, where $K$ is a field. Every such ideal arises as the complementary edge ideal $I_c(G)$ of a finite simple graph $G$. We describe the defining equations of the Rees algebra $\mathcal{R}(I_c(G))$ in terms of the combinatorics of $G$. If $G$ is a tree or a unicyclic graph whose unique induced cycle has length $3$ or $4$, we prove that $\mathcal{R}(I_c(G))$ is Koszul. We also determine the asymptotic depth of the powers of $I_c(G)$, proving that $\lim_{k \to \infty}\text{depth}\, S/I_c(G)^k=b(G)$, where $b(G)$ is the number of bipartite connected components of $G$. Finally, we show that the index of depth stability of $I_c(G)$ is at most $n-2$, and equality holds when $G$ is a path graph.

math.AC↗

arXiv2025

Complementary edge ideals

Let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$ and $I\subset S$ be a squarefree monomial ideal generated in degree $n-2$. Motivated by the remarkable behavior of the powers of $I$ when $I$ admits a linear resolution, as established in [11], in this work we investigate the algebraic and homological properties of $I$ and its powers. To this end, we introduce the complementary edge ideal of a finite simple graph $G$ as the ideal $$I_c(G)=((x_1\cdots x_n)/(x_ix_j):\{i,j\}\in E(G)) $$ of $S$, where $V(G)=\{1,\ldots,n\}$ and $E(G)$ is the edge set of $G$. By interpreting any squarefree monomial ideal $I$ generated in degree $n-2$ as the complementary edge ideal of a graph $G$, we establish a correspondence between algebraic invariants of $I$ and combinatorial properties of $G$. More precisely, we characterize sequentially Cohen-Macaulay, Cohen-Macaulay, Gorenstein, nearly Gorenstein and matroidal complementary edge ideals. Moreover, we determine the regularity of powers of $I$ in terms of combinatorial invariants of the graph $G$ and obtain that $I^k$ has linear resolution or linear quotients for some $k$ (equivalently for all $k\geq 1$) if and only if $G$ has only one connected component with at least two vertices.

math.AC↗

arXiv2025

Stanley-Reisner ideals with linear powers

Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees $d$, does there exist a uniform combinatorial characterization of all squarefree monomial ideals in $S$ having $d$-linear resolutions? (ii) For which degrees $d$, does having a linear resolution coincide with having linear powers for all squarefree monomial ideals of $S$ generated in degree $d$? Let $\mathcal{I}_{n,d}(K)$ denote the class of squarefree monomial ideals of $S$ having a $d$-linear resolution. Our main result establishes the equivalence of the following conditions: (a) Any squarefree monomial ideal $I$ in $S$ generated in degree $d$ has a linear resolution, if and only if, $I$ has linear powers. (b) $\mathcal{I}_{n,d}(K)$ is independent of the base field $K$. (c) $d\in\{0,1,2,n{-}2,n{-}1,n\}$. In each of these degrees, we show that a squarefree monomial ideal has a linear resolution if and only if all of its powers admit linear quotients, and we combinatorially classify such ideals. In contrast, for each degree $3\le d\le n{-}3$, we construct fully-supported squarefree monomial ideals $I$ and $J$ in $S$ generated in degree $d$ such that the linear resolution property of $I$ depends on the choice of the base field, $J$ has a linear resolution and $J^2$ does not have a linear resolution.

math.AC↗

arXiv2025

Monomial ideals whose all matching powers are Cohen-Macaulay

In the present paper, we aim to classify monomial ideals whose all matching powers are Cohen-Macaulay. We especially focus our attention on edge ideals. The Cohen-Macaulayness of the last matching power of an edge ideal is characterized, providing an algebraic analogue of the famous Tutte theorem regarding graphs having a perfect matching. For chordal graphs, very well-covered graphs and Cameron-Walker graphs, we completely solve our problem.

math.AC↗

arXiv2025

Algebraic study on permutation graphs

Let $G$ be a permutation graph. We show that $G$ is Cohen-Macaulay if and only if $G$ is unmixed and vertex decomposable. When this is the case, we obtain a combinatorial description for the $a$-invariant of $G$. Moreover, we characterize the Gorenstein permutation graphs.

math.AC↗

arXiv2025

Symbolic powers of polymatroidal ideals

In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal $I$, we conjecture that every symbolic power $I^{(k)}$ is componentwise linear and $$ \text{reg}\,I^{(k)}=\text{reg}\,I^k $$ for all $k \ge 1$. We prove that $\text{reg}\,I^{(k)}\ge\text{reg}\,I^k$ for all $k \ge 1$ when $I$ has no embedded associated primes, for instance if $I$ is a matroidal ideal. Moreover, we establish a criterion on the symbolic Rees algebra $\mathcal{R}_s(I)$ of a monomial ideal of minimal intersection type which guarantees that every symbolic power $I^{(k)}$ has linear quotients and, hence, is componentwise linear for all $k\ge1$. By applying our criterion to squarefree Veronese ideals and certain matching-matroidal ideals, we verify both conjectures for these families. We establish the Conforti-Cornuéjols conjecture for any matroidal ideal, and we show that a matroidal ideal is packed if and only if it is the product of monomial prime ideals with pairwise disjoint supports. Furthermore, we identify several classes of non-squarefree polymatroidal ideals for which the ordinary and symbolic powers coincide. Hence, we confirm our conjectures for transversal polymatroidal ideals and principal Borel ideals. Finally, we verify our conjectures for all polymatroidal ideals either generated in small degrees or in a small number of variables.

math.AC↗

arXiv2024

Sortable simplicial complexes and their associated toric rings

Let $Γ$ be a $d$-flag sortable simplicial complex. We consider the toric ring $R_Γ=K[{\bf x}_Ft:F\in Γ]$ and the Rees algebra of the facet ideals $I(Γ^{[i]})$ of pure skeletons of $Γ$. We show that these algebras are Koszul, normal Cohen-Macaulay domains. Moreover, we study the Gorenstein property, the canonical module, and the $a$-invariant of the normal domain $R_Γ$ by investigating its divisor class group. Finally, it is shown that any $d$-flag sortable simplicial complex is vertex decomposable, which provides a characterization of the Cohen-Macaulay property of such complexes.

math.AC↗

arXiv2024

Componentwise linear symbolic powers of edge ideals and Minh's conjecture

In this paper, we study the componentwise linearity of symbolic powers of edge ideals. We propose the conjecture that all symbolic powers of the edge ideal of a cochordal graph are componentwise linear. This conjecture is verified for some families of cochordal graphs, including complements of block graphs and complements of proper interval graphs. As a corollary, Minh's conjecture is established for such families. Moreover, we show that $I(G)^{(2)}$ is componentwise linear, for any cochordal graph $G$.

math.AC↗

arXiv2024

On the Rees algebra and the conductor of an ideal

For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the symmetric algebra of $I$ by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined.

math.AC↗

arXiv2024

Binomial ideals attached to finite collections of cells

We consider the ideal of inner $2$-minors $I_{\mathcal{P}}$ of a finite set of cells $\mathcal{P}$, which we call the cell ideal of $\mathcal{P}$. A nice interpretation for the height of an unmixed ideal $I_{\mathcal{P}}$, in terms of the number of cells of $\mathcal{P}$ is given. Moreover, the coordinate rings of cell ideals with isolated singularities are determined.

math.AC↗

arXiv2024

The divisor class group of a discrete polymatroid

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

math.AC↗

arXiv2023

Ideals and their Fitting ideals

For an ideal $I$ in a Noetherian ring $R$, the Fitting ideals $\textrm{Fitt}_j(I)$ are studied. We discuss the question of when $\textrm{Fitt}_j(I)=I$ or $\sqrt{\textrm{Fitt}_j(I)}=\sqrt{I}$ for some $j$. A classical case is the Hilbert-Burch theorem when $j=1$ and $I$ is a perfect ideal of grade $2$ in a local ring.

math.AC↗

arXiv2023

Normal Rees algebras arising from vertex decomposable simplicial complexes

We show that for a vertex decomposable simplicial complex $Δ$, the Rees algebra of $I_{Δ^{\vee}}$ is a normal Cohen-Macaulay domain. As consequences, we show that any squarefree weakly polymatroidal ideal is normal and we obtain normal ideals among several interesting families of monomial ideals such as cover ideals of graphs and edge ideals of hypergraphs. Moreover, based on a construction on simplicial complexes given by Biermann and Van Tuyl [2], we present families of normal ideals attached to any squarefree monomial ideal.

math.AC↗

arXiv2023

Toric rings attached to simplicial complexes

We consider standard graded toric rings $R_Δ$ whose generators correspond to the faces of a simplicial complex $Δ$. When $R_Δ$ is normal, it is shown that its divisor class group is free. For a flag complex $Δ$ which is the clique complex of a perfect graph, a nice description for the class group and the canonical module of $R_Δ$ in terms of the minimal vertex covers of the graph is given. Moreover, for a quasi-forest simplicial complex a quadratic Gröbner basis for the defining ideal of $R_Δ$ is presented. Using this fact we give combinatorial descriptions for the $a$-invariant and the Gorenstein property of $R_Δ$.

math.AC↗

arXiv2023

The eventual shape of the Betti table of $\mathfrak{m}^kM$

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $ \mathfrak{m}$ be the graded maximal ideal of $S$. It is known that for a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, the module $ \mathfrak{m}^kM$ is componentwise linear. For large $k$ we describe the pattern of the Betti table of $ \mathfrak{m}^kM$ when $\mathrm{char}(K)=0$ and $M$ is a submodule of a finitely generated graded free $S$-module. Moreover, we show that for any $k\gg 0$, $ \mathfrak{m}^kI$ has linear quotients if $I$ is a monomial ideal.

math.AC↗