On the arithmetical rank of certain monomial ideals

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 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.

The arithmetical rank of the edge ideals of graphs with whiskers

We consider the edge ideals of large classes of graphs with whiskers and for these ideals we prove that the arithmetical rank is equal to the big height. Then we extend these results to other classes of squarefree monomial ideals, generated in any degree, proving that the same equality holds.

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.

Tameness of Local cohomology of monomial ideals with respect to monomial prime ideals

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.

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.

On the arithmetical rank of the edge ideals of forests

We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.

$p-$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…

On the stable set of associated prime ideals of a monomial ideal

It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.

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.

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。

Monomial Difference Ideals

In this paper, basic properties of monomial difference ideals are studied. We prove the finitely generated property of well-mixed difference ideals generated by monomials. Furthermore, a finite prime decomposition of radical well-mixed monomial difference ideals is given. As a consequence, we prove that every strictly ascending chain of radical well-mixed monomial difference ideals in a difference polynomial ring is finite, which answers a question raised by E. Hrushovski in the monomial case. Moreover, the Alexander Duality for monomial ideals is generalized to the monomial case.

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 $I_G$.

The combinatorial calculation of algebraic invariants of a monomial ideal

We introduce the combinatorial Lyubeznik resolution of monomial ideals. We prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As an application, we give a combinatorial method to determine if an ideal is a Lyubeznik ideal. Furthermore, the minimality of the Lyubeznik resolution is characterized and we classify all the Lyubeznik symbols using combinatorial criteria. We get a combinatorial expression for the projective dimension, the length of Lyubeznik, and the arithmetical rank of a monomial ideal. We define the Lyubeznik totally ideals as those ideals that yield a mi…

The arithmetical rank of the edge ideals of graphs with pairwise disjoint cycles

We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.

Betti numbers of monomial ideals in four variables

We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in 4 variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in 4 variables.

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…

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.

Monomial ideals with linear quotients whose Taylor resolutions are minimal

We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal Taylor resolutions.

Koszul同调中的单项式闭链

Monomial Cycles in Koszul Homology

该研究探讨单项式环上Koszul同调的单项式闭链，提出边缘理想刻画闭链为边缘链的条件，推导Golod单项式理想的判定条件并完成多类Golod理想分类，还证明具线性商的一类理想存在单项式闭链构成的同调基。

Linear resolutions and quasi-linearity of monomial ideals

We introduce the concept of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and identify all the quasi-linear quadratic monomial ideals. We define a strongly linear monomial for a monomial ideal $I$ and prove that if $\mathbf{u}$ is a strongly linear monomial over $I$ then $I$ has a linear resolution (resp: is quasi-linear) if and only if $I+\mathbf{u}\mathfrak{p}$ has a linear resolution (resp: is quasi-linear). Here $\mathfrak{p}$ is any monomial prime ideal.

Minimal generating sets for the first syzygies of a monomial ideal

Two minimal generating sets of the first syzygies of a monomial ideal are produced, given the minimal generating set of the ideal.

Invariants of some classes of monomial ideals

In this thesis we are interested in describing some homological invariants of certain classes of monomial ideals. We will pay attention to the squarefree and non-squarefree lexsegment ideals.

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…

Stanley depth and size of a monomial ideal

Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by $1$ is a lower bound for its depth. We show that the size is also a lower bound for its Stanley depth. Applying Alexander duality we obtain upper bounds for the regularity and Stanley regularity of squarefree monomial ideals.

Monomial ideals of minimal depth and trivial modifications

Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's conjecture holds for a square free monomial ideal then it holds for all its trivial modifications.

The fiber cone of a monomial ideal in two variables

We determine in an explicit way the depth of the fiber cone and its relation ideal for classes of monomial ideals in two variables. These classes include concave and convex ideals as well as symmetric ideals.

关于单项式理想的共持续性质与近共持续性质

On the copersistence property and nearly copersistence property of monomial ideals

该论文聚焦数学交换代数领域，针对单项式理想展开研究，主要探究满足共持续性质或近共持续性质的这类单项式理想的相关特性。

Stanley depth of monomial ideals in three variables

We show that $\depth(S/I)=0$ if and only if $\sdepth(S/I)=0$, where $I\subset S=K[x_1,...,x_n]$ is a monomial ideal. We give an algorithm to compute the Stanley depth of $S/I$, where $I\subset S=K[x_1,x_2,x_3]$ is a monomial ideal. Also, we prove that a monomial ideal $I\subset K[x_1,x_2,x_3]$ minimally generated by three monomials has $\sdepth(I)=2$.

Monomial ideals whose depth function has any given number of strict local maxima

We construct monomial ideals with the property that their depth function has any given number of strict local maxima.

Expansions of monomial ideals and multigraded modules

We introduce an exact functor defined on multigraded modules which we call the expansion functor and study its homological properties. The expansion functor applied to a monomial ideal amounts to substitute the variables by monomial prime ideals and to apply this substitution to the generators of the ideal. This operation naturally occurs in various combinatorial contexts.

Monomial ideals under ideal operations

In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decompos…

A note on monomial ideals

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximal height of its minimal primes.

Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal

Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of $I$. We prove that there exists a unique minimal monomial reduction ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of $J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg 0$. We determine the structure of the reduced fiber ring $\mathcal{F}(J)_{\red}$ of $J$ and show that $\mathcal{F}(J)_{\red}$ is isomo…

Monomial ideals with large projective dimension

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S for which the quotient S/M has projective dimension n.

Colon structure of associated primes of monomial ideals

We find an explicit expression of the associated primes of monomial ideals as a colon by an element $v$, using the unique irredundant irreducible decomposition whose irreducible components are monomial ideals (Theorem 3.1). An algorithm to compute $v$ is given using Macaulay2 (Section 7). For squarefree monomial ideals the problem is related to the combinatorics of the underlying clutter or graph (Proposition 4.3). For ideals of Borel type the monomial $f$ takes a simpler form (Proposition 5.2). The authors classify when $f$ is unique (Proposition 6.2).

On monomial ideals whose Lyubeznik resolution is minimal

For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik ideal. In this paper, we characterize the Lyubeznik ideals, and we discover some classes of Lyubeznik ideals.

Decompositions of Betti diagrams of powers of monomial ideals: A stability conjecture

We state a conjecture on the stability of Betti diagrams of powers of monomial ideals.

Stanley depth of complete intersection monomial ideals

We compute the Stanley depth of irreducible monomial ideals and we show that the Stanley depth of a monomial complete intersection ideal is the same as the Stanley depth of it's radical. Also, we give some bounds for the Stanley depth of a monomial complete intersection ideal.

Simplicial affine semigroups with monomial minimal reduction ideals

We characterize when the monomial maximal ideal of a simplicial affine semigroup ring has a monomial minimal reduction. When this is the case, we study the Cohen-Macaulay and Gorenstein properties of the associated graded ring and provide several bounds for the reduction number with respect to the monomial minimal reduction.

Upper bounds of depth of monomial ideals

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds of $\depth_SI/J$.

Poset Resolutions of Monomial Ideals

We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $η:P\lra\mbb{N}^n$ a sequence of multigraded modules and maps.

Stanley depth of complete intersection monomial ideals and upper-discrete partitions

Let $I$ be an $m$-generated complete intersection monomial ideal in $S=K[x_1,...,x_n]$. We show that the Stanley depth of $I$ is $n-\floor{\frac{m}{2}}$. We also study the upper-discrete structure for monomial ideals and prove that if $I$ is a squarefree monomial ideal minimally generated by 3 elements, then the Stanley depth of $I$ is $n-1$.

A generalized Hochster's formula for local cohomologies of monomial ideals

The Hilbert series of local cohomologies for monomial ideals, which are not necessarily square-free, is established. As applications, we give a sharp lower bound of the non-vanishing degree of local cohomologies and also a sharp lower bound of the positive integer k of k-Buchsbaumness for generalized Cohen-Macaulay monomial ideals.

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.

Transversal Intersection of Monomial Ideals

In this paper we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon.

On the behavior of the size of a monomial ideal

In this paper we study the behavior of the size of a monomial ideal under polarization and under generic deformations. As an application, we extend a result relating the size and the Stanley depth of a squarefree monomial ideal obtained by Herzog, Popescu and Vladoiu, together with a parallel result obtained by Tang.

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.

On complete monomial ideals

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Classes of sequentially Cohen-Macaulay squarefree monomial ideals

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

具有正则商的单项式理想与某些边环

Monomial ideals with regular quotients and some edge rings

本文引入并研究了具有正则商的单项式理想作为线性商的推广，并据此计算出各类边环的 toric ideal 的 Betti 数。

Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals

Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.

单项式理想与最小广义巴里-马奇亚解析

Monomial ideals with minimal generalized Barile-Macchia resolutions

研究识别了几类具有最小广义巴里-马奇亚解析的单项式理想，包括通用单项式理想、具有线性商的单项式理想及超树的边理想，并刻画了具有桥友边理想的连通单环图。

单项完全交理想的商的Stanley深度

Stanley depth of quotient of monomial complete intersection ideals

本文计算了完全交单项式理想之商在特殊情形下的Stanley深度，给出了一般情形下的精确界，并证明了该情形满足Stanley猜想。