On the arithmetical rank of the edge ideals of forests

The arithmetical rank of a special class of monomial ideals

We give an affirmative answer to a question due to J. He and A. Van Tuyl, proving that the arithmetical rank of a special monomial ideal equals to the projective dimension of corresponding quotient module.

Arithmetical rank of binomial ideals

In this paper, we investigate the arithmetical rank of a binomial ideal $J$. We provide lower bounds for the binomial arithmetical rank and the $J$-complete arithmetical rank of $J$. Special attention is paid to the case where $J$ is the binomial edge ideal of a graph. We compute the arithmetical rank of such an ideal in various cases.

Arithmetical rank of Cohen-Macaulay squarefree monomial ideals of height two

In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.

Arithmetical ranks of Stanley-Reisner ideals of simplicial complexes with a cone

When a cone is added to a simplicial complex $Δ$ over one of its faces, we investigate the relation between the arithmetical ranks of the Stanley-Reisner ideals of the original simplicial complex and the new simplicial complex $Δ'$. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of $Δ'$ equals the projective dimension of the Stanley-Reisner ring of $Δ'$ if the corresponding equality holds for $Δ$.

On the arithmetical rank of an intersection of ideals

We determine sets of elements which, under certain conditions, generate an intersection of ideals up to radical.

Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four

Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $μ(I) \leq 5$; (2) $\arithdeg I \leq 4$.

Arithmetical rank of toric ideals associated to graphs

Let $I_{G} \subset K[x_{1},...,x_{m}]$ be the toric ideal associated to a finite graph $G$. In this paper we study the binomial arithmetical rank and the $G$-homogeneous arithmetical rank of $I_G$ in 2 cases: $G$ is bipartite, $I_G$ is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the $G$-arithmetical rank coincide with the minimal number of generators of $I_G$.

On the binomial arithmetical rank of lattice ideals

To any lattice $L \subset \mathbb{Z}^{m}$ one can associate the lattice ideal $I_{L} \subset K[x_{1},...,x_{m}]$. This paper concerns the study of the relation between the binomial arithmetical rank and the minimal number of generators of $I_{L}$. We provide lower bounds for the binomial arithmetical rank and the $\mathcal{A}$-homogeneous arithmetical rank of $I_{L}$. Furthermore, in certain cases we show that the binomial arithmetical rank equals the minimal number of generators of $I_{L}$. Finally we consider a class of determinantal lattice ideals and study some algebraic properties of them…

Specializations of multigradings and the arithmetical rank of lattice ideals

In this article we study specializations of multigradings and apply them to the problem of the computation of the arithmetical rank of a lattice ideal $I_{L_{\mathcal{G}}} \subset K[x_{1},...,x_{n}]$. The arithmetical rank of $I_{L_{\mathcal{G}}}$ equals the $\mathcal{F}$-homogeneous arithmetical rank of $I_{L_{\mathcal{G}}}$, for an appropriate specialization $\mathcal{F}$ of $\mathcal{G}$. To the lattice ideal $I_{L_{\mathcal{G}}}$ and every specialization $\mathcal{F}$ of $\mathcal{G}$ we associate a simplicial complex. We prove that combinatorial invariants of the simplicial complex provid…

Equations of 2-linear ideals and arithmetical rank

In this paper we consider reduced homogeneous ideals $\Jcal\subset S$ of a polynomial ring $S$, having a 2-linear resolution. 1. We study systems of generators of $\Jcal\subset S$. 2. We compute the arithmetical rank for a large class of projective curves having a 2-linear resolution. 3. We show that the fiber cone $\proj \Fcal(I_{\Lcal})$ of a lattice ideal $I_{\Lcal}$ of codimension two is a set theoretical complete intersection.

Arithmetical rank of strings and cycles

Let $R$ be a polynomial ring over a field $K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $H(I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $H(I)$ is a string or a cycle hypergraph.

A note on the edge ideals of Ferrers graphs

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

$p-$Ferrer diagram, $p-$linear ideals and arithmetical rank

In this paper we introduce $p-$Ferrer diagram, note that $1-$ Ferrer diagram are the usual Ferrer diagrams or Ferrer board, and corresponds to planar partitions. To any $p-$Ferrer diagram we associate a $p-$Ferrer ideal. We prove that $p-$Ferrer ideal have Castelnuovo mumford regularity $p+1$. We also study Betti numbers, minimal resolutions of $p-$Ferrer ideals. Every $p-$Ferrer ideal is $p-$joined ideals in a sense defined in a fortcoming paper \cite{m2}, which extends the notion of linearly joined ideals introduced and developped in the papers \cite{bm2}, \cite{bm4},\cite{eghp} and \cite{m1…

Simplicial ideals, 2-linear ideals and arithmetical rank

In the first part of this paper we study scrollers and linearly joined varieties. A particular class of varieties, of important interest in classical Geometry are Cohen--Macaulay varieties of minimal degree. They appear naturally studying the fiber cone of of a codimension two toric ideals. Let $I\subset S$ be an ideal defining a linearly joined arrangement of varieties: - We compute the depth, and the cohomological dimension. is the connectedness dimension. - We characterize sets of generators of $I$, and give an effective algorithm to find equations, as an application we compute arithmetical…

广义二项式边理想的算术秩与上同调维数

Arithmetical rank and cohomological dimension of generalized binomial edge ideals

本文研究连通简单图对应的广义二项式边理想，给出其上同调维数下界，刻画上同调完全交条件，并在多种情形下证明 $J_2(G)$ 的算术秩等于商环投射维数。

Cohomological dimension and arithmetical rank of some determinantal ideals

Let $M$ be a $(2 \times n)$ non-generic matrix of linear forms in a polynomial ring. For large classes of such matrices, we compute the cohomological dimension (cd) and the arithmetical rank (ara) of the ideal $I_2(M)$ generated by the $2$-minors of $M$. Over an algebraically closed field, any $(2 \times n)$-matrix of linear forms can be written in the Kronecker-Weierstrass normal form, as a concatenation of scroll, Jordan and nilpotent blocks. Bădescu and Valla computed $\mathrm{ara}(I_2(M))$ when $M$ is a concatenation of scroll blocks. In this case we compute $\mathrm{cd}(I_2(M))$ and exten…

Arithmetical rank of the cyclic and bicyclic graphs

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.

Some characterizations of strongly irreducible submodules in arithmetical and Noetherian modules

The purpose of the present paper is to prove some properties of the strongly irreducible submodules in the arithmetical and Noetherian modules over a commutative ring. The relationship among the families of strongly irreducible submodules, irreducible submodules, prime submodules and primal submodules is proved. Also, several new characterizations of the arithmetical modules are given. In the case when $R$ is Noetherian and $M$ is finitely generated, several characterizations of strongly irreducible submodules are included. Among other things, it is shown that when $N$ is a submodule of $M$ su…

The minimal cellular resolutions of the edge ideals of forests

We present an explicit construction of a minimal cellular resolution for the edge ideals of forests, based on discrete Morse theory. In particular, the generators of the free modules are subsets of the generators of the modules in the Lyubeznik resolution. This procedure allows to ease the computation of the graded Betti numbers and the projective dimension.

Squarefree powers of edge ideals of forests

Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.

The Betti numbers of forests

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs.

Optimal Betti numbers of forest ideals

We prove a tight lower bound on the Betti numbers of tree and forest ideals and a tight upper bound on certain graded Betti numbers of squarefree monomial ideals.

On ideals generated by monomials and one binomial

We determine, in a polynomial ring over a field, the arithmetical rank of certain ideals generated by a set of monomials and one binomial.

Projective dimensions of the edge ideals of forests

This paper is currently undergoing revision.

Projections of cones and the arithmetical rank of toric varieties

Let $I_M$ and $I_N$ be defining ideals of toric varieties such that $I_M$ is a projection of $I_N$, i.e. $I_N \subseteq I_M$. We give necessary and sufficient conditions for the equality $I_M=rad(I_N+(f_1,...,f_s))$, where $f_1,...,f_s$ belong to $I_M$. Also a method for finding toric varieties which are set-theoretic complete intersection is given. Finally we apply our method in the computation of the arithmetical rank of certain toric varieties and provide the defining equations of the above toric varieties.

Betti Numbers of Graph Ideals

In this thesis we investigate certain types of monomial ideals of polynomial rings over fields. We are interested in minimal free resolutions of these ideals (or equivalently the quotients of the polynomial ring by the ideals) considered as modules over the polynomial ring. There is no simple method of finding such resolutions but in the case of Stanley-Reisner ideals Hochster's formula and its variants provide a way to compute the Betti numbers of these resolutions. Even with these formulae it is not in general possible to find especially explicit or useful descriptions of the Betti numbers. …

Regularity of powers of edge ideals of Cohen-Macaulay weighted oriented forests

In this paper, we explicitly give combinatorial formulas for the regularity of powers of edge ideals, $\reg(I(D)^k)$, of weighted oriented unmixed forests $D$ whose leaves are sinks ($V^+(D)$ are sinks). This combinatorial formula is a piecewise linear function of $k$, for $k \geq 1$.

Projective dimension and regularity of powers of edge ideals of vertex-weighted rooted forests

In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also give some examples to show that these formulas are related to direction selection and the assumptions about "rooted" forest such that $w(x)\geq 2$ if $d(x)\neq 1$ cannot be dropped.

匹配幂为线性的森林

Forests whose matching powers are linear

本文分类了边理想的某个匹配幂具有线性分解性质的赋权有向森林。

图的幂的边理想的正则性

Regularity of edge ideals of powers of graphs

本文研究图的幂的边理想的正则性，证明了森林的幂的边理想正则性弱递减，并计算了圈的幂的边理想正则性。

Lattice Complements and the Subadditivity of Syzygies of Simplicial Forests

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.

无平方幂的Cohen-Macaulay simplicial森林

Square-free powers of Cohen-Macaulay simplicial forests

本文研究了Cohen-Macaulay simplicial森林的无平方幂性质，证明其在所有k≥1时均为Cohen-Macaulay，并引入特殊叶的概念，给出深度函数的显式公式。

Schmitt-Vogel type lemma for reductions

The lemma given by Schmitt and Vogel is an important tool in the study of arithmetical rank of squarefree monomial ideals. In this paper, we give a Schmitt-Vogel type lemma for reductions as an analogous result.

Regularity of powers of forests and cycles

Let G be a graph and let I = I(G) be its edge ideal. In this paper, when G is a forest or a cycle, we explicitly compute the regularity of I^s for all s > 0. In particular, for these classes of graphs, we provide the asymptotic linear function reg(I^s) as s > 0, and the initial value of s starting from which reg(I^s) attains its linear form. We also give new bounds on the regularity of I when G contains a Hamiltonian path and when G is a Hamiltonian graph.

Pretty clean monomial ideals and linear quotients

We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal $I$ has linear quotients, then the squarefree part of $I$ and each component of $I$ as well as $\mm I$ have linear quotients, where $\mm$ is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly. As a generalization of the facet ideal of a forest, we define monomial ideals of forest type and show that…

Depths of Powers of the Edge Ideal of a Tree

Lower bounds are given for the depths of R/I^t for t at least one when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.

二项式边理想的解析扩张

Analytic spread of binomial edge ideals

本文研究有限简单图的二项式边理想的解析扩张，给出一般紧界，并针对闭图和伪森林等特殊图族用组合与凸几何方法计算精确值。

Linear strands of edge ideals of multipartite uniform clutters

We construct the first linear strand of the minimal free resolutions of edge ideals of $d$-partite $d$-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing the known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the co…

Non-vanishingness of Betti numbers of edge ideals

Given finite simple graph one can associate the edge ideal. In this paper we discuss the non-vanishingness of the graded Betti numbers of edge ideals in terms of the original graph. In particular, we give a necessary and sufficient condition for a chordal graph on which the graded Betti number does not vanish and characterize the graded Betti number for a forest. Moreover we characterize the projective dimension for a chordal graph.

图的闭邻域理想的Castelnuovo-Mumford正则度

Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph

该研究围绕图的闭邻域理想的同调不变量展开，证明了森林情形下其Castelnuovo-Mumford正则度等于匹配数（证实Sharifan与Moradi的猜想），给出任意图下该正则度的匹配数下界，还证明含单纯顶点时理想有Betti分裂，且森林与单圈图的投射维数也以匹配数为下界。

Regularity and projective dimension of edge ideal of $C_5$-free vertex decomposable graphs

In this paper, we explain the regularity, projective dimension and depth of edge ideal of some classes of graphs in terms of invariants of graphs. We show that for a $C_5$-free vertex decomposable graph $G$, $\T{reg}(R/I(G))= c_G$, where $c_G$ is the maximum number of 3-disjoint edges in $G$. Moreover for this class of graphs we characterize $\T{pd}(R/I(G))$ and $\T{depth}(R/I(G))$. As a corollary we describe these invariants in forests and sequentially Cohen-Macaulay bipartite graphs.

Projective dimension and regularity of edge ideal of some weighted oriented graphs

In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. As some consequences, we give some exact formulas for the depth of these ideals.

单项式理想的匹配幂与赋权有向图的边理想

Matching powers of monomial ideals and edge ideals of weighted oriented graphs

本文引入单项式理想的匹配幂概念，研究了单项式理想及赋权有向图边理想匹配幂的深度与正则性，并证明了二次单项式理想最后非零幂的多拟阵性及赋权有向森林边理想具有线性分解的条件。

Algebraic properties of edge ideals via combinatorial topology

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edges ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We app…

Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equ…

Critical ideals of trees

Given a graph $G=(V, E)$, its generalized Laplacian matrix is given by \[ L(G,X_G)_{u,v}= \begin{cases} x_u&\text{if }u=v,\\ -m_{uv}&\text{if }u\neq v, \end{cases} \] where $X_G=\{x_u\, | \, u\in V(G)\}$ is a set of indeterminates and $m_{uv}$ is the number of edges between $u$ and $v$. The $j$-critical ideal of $G$ is the determinantal ideal generated by the minors of size $j$ of $L(G, X)$. A $2$-matching of $G$ is a subset $\mathcal{M}$ of its edges such that every vertex of $G$ has at most two incident edges in $\mathcal{M}$. We give a combinatorial description of a set of generators of the…

Koszul homology of quotients by edge ideals

We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.

Tetrahedral Curves

A tetrahedral curve is a space curve whose defining ideal is an intersection of powers of monomial prime ideals of height two. It is supported on a tetrahedral configuration of lines. Schwartau described when certain such curves are ACM, namely he restricted to curves supported on a certain four of the six lines. We consider the general situation. We first show that starting with an arbitrary tetrahedral curve, there is a particular reduction that produces a smaller tetrahedral curve and preserves the even liaison class. We call the curves that are minimal with respect to this reduction S-mini…

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally general…

Dirac's theorem and multigraded syzygies

Let $G$ be a simple finite graph. A famous theorem of Dirac says that $G$ is chordal if and only if $G$ admits a perfect elimination order. It is known by Fröberg that the edge ideal $I(G)$ of $G$ has a linear resolution if and only if the complementary graph $G^c$ of $G$ is chordal. In this article, we discuss some algebraic consequences of Dirac's theorem in the theory of homological shift ideals of edge ideals. Recall that if $I$ is a monomial ideal, $\mbox{HS}_k(I)$ is the monomial ideal generated by the $k$th multigraded shifts of $I$. We prove that $\mbox{HS}_1(I)$ has linear quotients, …