Mircea Cimpoeas
We introduce a new class of monomial ideals, called strong Borel type ideals, and we compute the Mumford-Castelnouvo regularity for principal strong Borel type ideals. Also, we describe the d-fixed ideals generated by powers of variables and we compute their regularity.
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This paper gives explicit formulas for the reduction number and the Castelnuovo-Mumford regularity of projective monomial curves.
We survey a number of recent studies of the Castelnuovo-Mumford regularity of squarefree monomial ideals. Our focus is on bounds and exact values for the regularity in terms of combinatorial data from associated simplicial complexes and/or hypergraphs.
We show that the Castelnuovo--Mumford regularity of the canonical or a deficiency module of the quotient of a polynomial ring by a monomial ideal is bounded by its dimension.
We compare, for smooth monomial projective curves, the Castel- nuovo-Mumford regularity and the reduction number; we present an example where these two numbers differ. However, we show they coin- cide for a certain class of monomial curves. Furthermore, for smooth monomial curves we prove an inequality which is stronger than the one from the Eisenbud-Goto conjecture.
Regularity of linearly presented squarefree monomial ideals
本文证明了线性表示无平方单项式理想正则度的精确界，并据此肯定回答了 Dao–Takagi 关于满足 Serre $S_2$ 条件者上同调维数的问题。
In this paper, we extend a result of Eisenbud-Reeves-Totaro in the frame of ideals of Borel type. As a consequence, we obtain a linear upper bound for the regularity of a new class of ideals, called $\mathcal D$-fixed 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.
Let $I$ be a two-dimensional squarefree monomial ideal of a polynomial ring $S$. We evaluate the geometric regularity, $a_i$-invariants for $i\geq 1$ of the power $I^n$. It turns out they are all linear functions in $n$ from $n=2$. Moreover, it is proved $\mbox{g-reg}(S/I^n)=\reg(S/I^{(n)})$ for all $n\geq 1$.
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.
Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute the resurgence of ${\mathfrak p}$, the Waldschmidt constant of ${\mathfrak p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of ${\mathfrak p}$.
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.
Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves
本文构造了无限类非光滑非Cohen-Macaulay的k-Buchsbaum射影单项式曲线，给出其Macaulay化的生成元，并探讨正则度与k-Buchsbaum性的联系。
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.
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.
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.
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。
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.
Two minimal generating sets of the first syzygies of a monomial ideal are produced, given the minimal generating set of the ideal.
We determine a new technique which allows the computation of the arithmetical rank of certain 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.
On regularity and projective dimension of invariant chains of monomial ideals
该研究针对递增函数幺半群作用下不变的无限维多项式环理想的截断渐近行为猜想，证明单项式理想的投射维数最终呈线性、正则性渐近呈线性，为相关猜想提供了证据。
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.
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 regularity of monomial ideals and their integral closures
本文研究了单项式理想及其整数闭包的正则性，证明了闭包的正则性不高于原理想，并给出了正则性等于度数的充要条件。
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.
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。
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 $\sdepth(I)=2$.
We construct monomial ideals with the property that their depth function has any given number of strict local maxima.
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.
In this paper, we prove that a squarefree monomial ideal of height 2 whose quotient ring is Cohen-Macaulay is set-theoretic complete intersection.
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…
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.
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…
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.
Castelnuovo-Mumford regularity is a measure of algebraic complexity of an ideal. Regularity of monomial ideals can be investigated combinatorially. We use a simple graph decomposition and results from structural graph theory to prove, improve and generalize many of the known bounds on regularity of quadratic square-free 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).
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.
We state a conjecture on the stability of Betti diagrams of powers of 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.
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.
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 $\depth_SI/J$.
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.
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 $n-1$.
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.
In this paper we prove conditions for transversal intersection of monomial ideals and derive a simplicial characterization of this phenomenon.
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.
In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.
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.
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。
We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_Δ$ is the Stanley-Reisner ideal of a simplicial complex $Δ$, then $\reg(I^{(n)}) \leqslant δ(n-1) +b$ for all $n\geqslant 1$, where $δ= \lim\limits_{n\to\infty} \reg(I^{(n)})/n$, and $b = \max\{\reg(I_Γ) \mid Γ\text{ is a subcomplex of } Δ\text{ with } \F(Γ) \subseteq \F(Δ)\}$. This bound is sharp for any $n$. When $I = I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\reg(I^{(n)}) \leqslant 2n + \ordmatch(G)-1$, where $\ordmatch(G)$ is the ordered matching n…
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.
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。
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。
We study the deformation theory of quotients of polynomial rings by quadratic monomial ideals. More precisely we compute the first cotangent cohomology module of such rings. We also give a criterion for vanishing of second cotangent cohomology module.
Further results on monomial ideals of projective dimension one
本文证明了投射维数一的单项式理想等价于其极小生成元可排序使得连续商理想为主理想，并给出了此类理想中具有线性分解的平方自由单项式理想的部分分类。
We give explicit formulas for the Bhattacharya function of m-primary complete monomial ideals in two variables in terms of the vertices of the Newton polyhedra or in terms of the decompositions of the ideals as products of simple ideals.
We give a generalization of Hochster's formula for local cohomologies of square-free monomial ideals to monomial ideals, which are not necessarily square-free. Using this formula, we give combinatorial characterizations of generalized Cohen-Macaulay monomial ideals. We also give other applications of the generalized Hochster's formula.
Given a monomial ideal in a polynomial ring over a field, we define the LCM-dual of the given ideal. We show good properties of LCM-duals. Including the isomorphism between the special fiber of LCM-dual and the special fiber of given monomial ideal. We show the special fibers of LCM-duals of strongly stable ideals are normal Cohen-Macaulay Koszul domains. We provide an explicit describing of minimal free resolutions of LCM-duals of strongly stable ideals.
In this paper, we introduce techniques for producing normal square-free monomial ideals from old such ideals. These techniques are then used to investigate the normality of cover ideals under some graph operations. Square-free monomial ideals that come out as linear combinations of two normal ideals are shown to be not necessarily normal; under such a case we investigate the integral closedness of all powers of these ideals.
In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a min…
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