Susan Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-Tang, Susan Morey, Liana M. Sega, Sandra Spiroff
Let $I$ be a square-free monomial ideal $I$ of projective dimension one. Starting with the Taylor complex on the generators of $I^r$, we use Discrete Morse theory to describe a CW complex that supports a minimal free resolution of $I^r$. To do so, we concretely describe the acyclic matching on the faces of the Taylor complex.
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Morse resolutions of monomial ideals and Betti splittings
本文利用离散 Morse 理论与分裂技巧研究单项式理想的自由消解，证明剪枝消解对稳定理想和线性商理想具有极小性，并统一了 Eliahou-Kervaire 与 Herzog-Takayama 等经典构造，同时提出约化极小性研究的方法及幂理想剪枝消解变体。
The MorseResolutions package for Macaulay2
本文介绍了Macaulay2程序包MorseResolutions，用于处理Lyubeznik和Barile-Macchia两类Morse消解，并支持搜索给定单项式理想的最小Barile-Macchia消解。
We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen-Macaulay property. Indeed, we study the class of monomial ideals $I$, whose projective dimension is stable under monomial localizations at monomial prime ideals $\fp$, with $\height \fp\geq \pd S/I$. We study the relations between this property and other sorts of Cohen-Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.
This paper gives explicit formulas for the reduction number and the Castelnuovo-Mumford regularity of projective monomial curves.
Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in general, but we can slightly modify our algorithm in order to obtain a simplicial resolution. We also show that the Lyubeznik resolution fits into our pruning strategy. We finally use our methods to give a different approach to the theory of splitting of monomial ideals.
Nearly Gorenstein projective monomial curves of small codimension
本文研究小余维数情形下的近Gorenstein射影单项曲线，完整刻画了余维数为2和3的这类曲线的结构与分类。
Monomial ideals with large projective dimension
设S是任意域上的n元多项式环。对S中的每个单项式理想M，给出了S/M的全Betti数、分次Betti数和多重分次Betti数贝蒂数.我们还给出了S中商S/M具有投射维数n的所有单项式理想M的一个明确的刻画.
When are Morse resolutions polyhedral?
本文研究 Morse 匹配所得复形何时为多面体胞腔复形，证明至多四个极小生成元时总存在合适的极大匹配，并给出六生成元理想不存在多面体极小自由分解的反例。
The aim of this note is to exploit a new relationship between additive combinatorics and the geometry of monomial projective curves. We associate to a finite set of non-negative integers $A=\{a_1,\cdots, a_n\}$ a monomial projective curve $C_A\subset \mathbb P^{n-1}_{k}$ such that the Hilbert function of $C_A$ and the cardinalities of $sA:=\{a_{i_1}+\cdots+a_{i_s}\mid 1\le i_1\le \cdots \le i_s\le n\}$ agree. The singularities of $C_A$ determines the asymptotic behaviour of $|sA|$, equivalently the Hilbert polynomial of $C_A$, and the asymptotic structure of $sA$. We show that some additive in…
We study the projective closures of three important families of affine monomial curves in dimension $4$, namely the Backelin curve, the Bresinsky curve and the Arslan curve, in order to explore possible connections between syzygies and the arithmetic Cohen-Macaulay property.
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。
Resolutions of Monomial Ideals of Projective Dimension 1
我们证明了一个单项理想$I$有投射维数$\leq$ 1当且仅当$S/I$的最小自由分解是一棵树的图的支持。这是通过构建支持$S/I$解析的特定图形来完成的。我们还提供了拟树的一个新的刻画，并利用这个刻画对Herzog，Hibi和Zheng关于投射维数为1的单项理想用拟树刻画的一个结果给出了一个新的证明.
On complete intersection projective closures of monomial curves
研究四维仿射空间中单项式曲线射影闭包的完全交性质，给出其定义理想为完全交的刻画，还探讨其是否算术科恩 - 麦考利。
The projective dimension of sequentially Cohen-Macaulay monomial ideals
证明了序列Cohen-Macaulay无平方单项式理想的投射维数等于它的极小素数的最大高度（也称为大高度），或者等价地等于它的小平面复形的极小顶点覆盖的最大基数.这特别给出了一个公式的投射维数方面的理想，这些类的理想，这是已知的顺序科恩-麦考利：图树和单纯树，弦图和一些周期，弦杂波和图形，以及一些路径理想，仅举几例。由于极化保持了投射维数，我们的结果也给出了序列Cohen-Macaulay单项理想的投射维数.
Bounding the Projective Dimension of a Square-Free Monomial Ideal via Domination in Clutters
我们在杂波中引入边支配的概念，并使用它来提供任何无平方单项理想的投射维数的上界。然后，我们使用一个简单的递归来恢复一个公式的投射维数的单项式理想相关联的弦杂波，定义由Woodroofe在\cite{russ}。我们还研究了一个家庭的杂波相关的图，并表明，这些杂波弦当且仅当相关的图。最后，我们计算这些杂波的某些类别的支配参数。
In this paper our aim is twofold. First, we introduce the notion of star gluing of numerical semigroups and show that arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure are preserved under this gluing operation. We then give a condition on Gröbner basis of the defining ideal of an affine monomial curve which ensures that the Betti sequence of the affine curve is the same as the Betti sequence of its projective closure. We also study the effect of simple gluing on Betti sequences of the projective closure. Finally, we construct some numerical semigroups, using a g…
Cellular resolutions of powers of monomial ideals
理想的不同幂的不变量之间有许多联系。我们研究如何构造最小的决议，所有权力在一次使用的方法，从代数和多面体拓扑，重点是理想所产生的组合。在一个构造中，我们得到了由（n-2）维复形支持的n个顶点上的二部图的边理想的所有幂的胞腔分解。我们的主要结果是一个明确的最小细胞决议的所有权力的边缘理想的道路。这些胞腔复形是通过首先细分多面体复形，然后使用离散的莫尔斯理论修改它们来构造的。
We prove new characterizations based on Gröbner bases for the Cohen-Macaulay property of a projective monomial curve.
Some Ideals with Large Projective Dimension
对于域上多项式环中的理想$I$，$I$的单项式支撑是作为$I$的极小生成元集合中的项出现的单项式的集合。Craig Huneke提出了一个问题：单项式支持的大小是否是理想的投射维数的一个界。我们构造了一个例子来表明，如果变量的数量和程度的发电机是未指定的，投射维数的$I$增长至少指数的单项式支持的大小。我们构造的理想是由单项式和二项式生成的。
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。
Cellular resolutions of second powers of square-free monomial ideals with divisibility relations
本文研究了具有可除关系的平方自由单项理想二次幂的细胞自由 resolutions，利用离散 Morse 理论构造了特定理想下的最小 resolutions，并给出了二次幂的射影维数的精确界。
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。
Dependence of Betti Numbers on Characteristic
研究了单项理想的分次贝蒂数对基域特征的依赖性.我们描述的例子包括二分理想，斯坦利-Reisner理想的顶点可分解的复合体和理想与组件的线性决议。我们给出了一个描述的二部图，并使用离散莫尔斯理论，提供了一种方式来看待通过二分理想的任意单纯复形的同源性。我们还证明了有理数域上的单项理想的Betti表可以由任意域上的Betti表通过一系列的连续消去得到。
Seshadri stratifications and standard monomial theory
本文提出嵌入射影簇上的 Seshadri 分层，用以构造 Newton-Okounkov 单纯复形与平坦环面退化，并为标准单项式理论及 Schubert 簇中 LS 路径的几何解释提供统一框架。
Bouquets, vertex covers and edge ideals
本文给出了无平方因子的单项理想的大高度的一个新的组合刻划，从而给出了单项理想的投射维数的一个新的界。
Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero.
Projective Dimension of String and Cycle Hypergraphs
本文给出了一个闭公式和一个简单的算法来计算与弦超图或圈超图相关的无平方单项式理想的投射维数。作为应用，在这些理想中，我们刻画了所有的Cohen-Macaulay理想。
The arithmetical rank of a special class of monomial ideals
本文对J.He和A. Van Tuyl等证明了一个特殊的单项理想的算术秩等于相应的商模的投射维数。
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。
Hypergraphs with high projective dimension and 1-dimensional Hypergraphs
本文证明了一个与超图（对偶）相关联的无平方单项式理想$J$的投射维数等于$J$的最小生成元数减2的充分必要条件。我们还提供了一个有效的显式程序来计算$\mathcal{H}$的1维超图的投射维数时，每个连通分支包含最多一个循环。还包括计算投射维数的算法。这些结果的应用，它们包括，例如，计算的投射维数的单项理想，其相关的超图有一个生成费勒斯图。
Complete intersections in certain affine and projective monomial curves
设$k$是任意域，本文的目的是给出正整数族$\mathcal{A} = \{d_1,\ldots,d_n\}$，使得仿射单项曲线$\mathcal C = \{(t^{d_1},\ldots,\,t^{d_n}) \ | \ t \in k\} \subset \mathbb{A}_k^n$的复曲面理想$I_{\mathcal A}$或其射影闭包${\mathcal C^{\star}} \subset \mathbb{P}_k^n$的复曲面理想$I_{\mathcal A^{\star}}$是完全交.更准确地说，我们刻画了$I_{\mathcal A}$和$I_{\mathcal A^{\star}}$的完全相交性质，当：（a）$\mathcal{A}$是一个广义算术序列，（b）$\mathcal{A} \setminus \{d_n\}$是一个广义算术序列和$d_n \in \mathbb{Z}^+$，（c）$\mathcal{A}$由$(p,q)$ -Fibonacci序列的某些项组成，和（d）$\mathcal{A}$由$(p,q)$ -Lucas序列的某些项组成。本文中的结果是Bermejo等人[J. Symb. Comput. 42（2007）]、Bermejo和Garc 'ifa-Marco [J. Symb. Comput.（2014），出现]和一些新的结果，关于复…
Castelnuovo-Mumford regularity and reduction number of smooth monomial curves
我们比较，光滑单项射影曲线，卡斯特-诺沃-芒福德正则性和减少数，我们提出了一个例子，这两个数字不同。然而，我们证明了它们对某类单项曲线是coin-cide的.此外，对于光滑的单项曲线，我们证明了一个不等式，这是强于一个从艾森巴德-后藤猜想。
Betti Numbers of Graph Ideals
本文主要研究域上多项式环的单项理想。我们感兴趣的是这些理想（或等价的多项式环的理想的同分项）作为多项式环上的模的极小自由分解。没有简单的方法找到这样的决议，但在斯坦利-赖斯纳理想的情况下，Hochster的公式及其变种提供了一种方法来计算这些决议的贝蒂数。即使有了这些公式，一般也不可能找到贝蒂数的特别明确或有用的描述。然而，我们限制我们的注意力，这些理想所产生的平方自由单项式的程度2。这样做的目的是将这些理想与图联系起来。这就提供了代数对象（单项式理想）和组合对象（图）之间的联系。这种对应关系使我们能够定义图的新的数值不变量：相应图理想的贝蒂数和投射维数。我们发现明确的描述的贝蒂数和投射维数的周期和森林。在森林的情况下，我们找到了一种用子森林的贝蒂数来描述贝蒂数的方法。这也导致了一个森林的投影维数的描述方面的投射维数的子森林。证明了森林的投射维数可以用纯组合的术语来定义，从而给出了森林的一个新的组合数值不变量。
Finite subsets of projective space, and their ideals
设$\mathscr{A}$是射影空间中闭有理点的有限集，$\mathscr{I}$是$\mathscr{A}$的消失理想，$\mathscr{D}(\mathscr{A})$是不作为$\mathscr{I}$中元素的首单项式出现的那些单项式的指数集.我们表明，$\mathscr{A}$的大小等于$\mathscr{D}(\mathscr{A})$中包含的轴的数量。此外，我们提出了一个算法的Gröbner基的$\mathscr{I}(\mathscr{A})$的建设，因此也$\mathscr{D}(\mathscr{A})$。
We give a new proof of Hilbert's Syzygy Theorem for monomial ideals. In addition, we prove the following. If S=k[x_1,...,x_n] is a polynomial ring over a field, M is a squarefree monomial ideal in S, and each minimal generator of M has degree larger than i, then the projective dimension of S/M is at most n-i.
Inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally-backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals.
Bounds on the regularity and projective dimension of ideals associated to graphs
在本文中，我们给出了新的上界的正则性的边缘理想的决议是k-步骤的线性，令人惊讶的是，界的对数的变量的数量。我们也给出了各种界限的投射维数这样的理想，推广了其他最近的结果。通过Alexander对偶，我们的结果也适用于余维为2的非混合无平方项理想。我们也讨论和连接这些结果在交换代数更经典的主题。
Derivation module and the Hilbert-Kunz multiplicity of the co-ordinate ring of a projective monomial curve
针对参数对应等差数列的射影单项式曲线坐标环，本文显式构造其导子模的极小生成元，并给出 Hilbert–Kunz 重数公式。
Structural decomposition of monomial resolutions
本文利用两类基本理想的重分次贝蒂数来表示任意单项理想的重分次贝蒂数。这种分解具有多种应用。在某些具体情况下，我们用它来构造单项理想类的极小分解;在其他情况下，我们用它来计算投射维数。为了说明结构分解的有效性，我们给出了Charalambous的一个经典定理的新证明。
Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences
设$K$是一个无限域，$m_1,\ldots,m_n$是一个正整数的广义等差数列，即，存在$h, d, m_1 \in\mathbb{Z}^+$，使得对于所有$i \in \{2,\ldots,n\}$都有$m_i = h m_1 + (i-1)d$。考虑由$$x_1=s^{m_1}t^{m_n-m_1},\dots,x_{n-1}=s^{m_{n-1}}t^{m_n-m_{n-1}},x_n=s^{m_n},x_{n+1}=t^{m_n}.$$参数化定义的射影单项曲线$\mathcal C\subset \mathbb{P}^{n}_{K}$在这篇文章中，我们刻画了$\mathcal C$的齐次坐标环$K[\mathcal C]$的Cohen-Macaulay和Koszul性质.当$K[\mathcal C]$是Cohen-Macaulay型时，我们也得到了它的Cohen-Macaulay型的一个公式。此外，当$h$整除$d$时，我们得到了$\mathcal C$的消失理想关于度逆字典序的极小Gröbner基$\mathcal G$.从$\mathcal G$中我们导出了$K[\mathcal C]$的Castelnuovo-Mumford正则性、Hilbert级数和Hilbert函数的公式。
For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules Ext^j(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the S_n-invariant monomial ideals that have a linear free resolution, and also characterize those which are Cohen-Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and anoth…
Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated hypergraph is either a string or a cycle. We investigate the impact on the projective dimension when higher dimensional edges are removed. We prove that the higher dimensional edge either has no effect on the projective dimension or the projective dimension only goes up by one with the extra higher dimensional edge.
Monomial invariants in codimension two
定义了射影簇$Z$的单项不变量，它们是来自于$Z$的一般初始理想的不变量.利用这一概念，我们推广了Cook的一个结果：如果$Z$是余维2的整数簇，满足附加假设$s_Z=s_Γ,$，那么它的单项不变量是连通的.
Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four
设$I$是多项式环$S$的一个无平方根的单项理想.本文证明了当满足下列条件之一时，$I$的算术秩等于$S/I$的投射维数：（1）$μ(I) \leq 5$ ;（2）$\arithdeg I \leq 4$ .
Determinantal ideals and monomial curves in the three-dimensional space
我们证明了在仿射或射影三维空间中的每一条单项曲线的定义理想可以由三个二项方程集理论上定义，其中两个集理论上定义了一个由具有单项元素的$2\times 3$矩阵的2-未成年人生成的行列式理想。
Betti numbers of monomial ideals via facet covers
本文给出了单项理想在每个多重度上有非零贝蒂数的一个充分条件。在单纯森林的小面理想的情况下，这个条件成为一个必要条件，它使我们能够组合地刻画这种理想的贝蒂数，投射维数和正则性。我们的条件是表示在单纯复形的最小面覆盖。
Properties of LCM Lattices of Monomial Ideals
研究单项式理想LCM格的格性质，包括布尔格、模格等，并证明极小单项式理想在模格下的Cohen-Macaulay性质，以及项目维数与LCM格高度的关系。
This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of square-free monomial ideals of projective dimension one, by introducing a combinatorial construction of a family of (cubical) cell complexes whose 1-skeletons are powers of a graph that supports the resolution of the ideal.
Projective monomial curves associated to numerical semigroups with multiplicity $e$, width $e-1$, and embedding dimension $e-2$
本文研究重数为e、宽度为e-1、嵌入维数为e-2的数值半群S(e,m,n)所定义的射影单项曲线，得到其定义理想的Gröbner基，并刻画了曲线的Cohen–Macaulay与Gorenstein性质及正则度。
The combinatorial calculation of algebraic invariants of a monomial ideal
我们引入了单项理想的组合Lyubeznik分解。我们证明了这个决议是同构于通常的Lyubezbnik决议。作为应用，我们给出了一个判定理想是否为Lyubeznik理想的组合方法.此外，最小的Lyubeznik决议的特点，我们分类所有的Lyubeznik符号使用组合标准。我们得到了一个关于单项理想的投射维数、Lyubeznik的长度和算术秩的组合表达式。我们定义Lyubeznik完全理想为在任何全序下产生最小自由分解的理想。最后，我们证明了对于一类图，它们的边理想是Lyubeznik完全理想。
We study monomial cut ideals associated to a graph $G$, which are a monomial analogue of toric cut ideals as introduced by Sturmfels and Sullivant. Primary decompositions, projective dimensions, and Castelnuovo-Mumford regularities are investigated if the graph can be decomposed as $0$-clique sums and disjoint union of subgraphs. The total Betti numbers of a cycle are computed. Moreover, we classify all Freiman ideals among monomial cut ideals.
Algebraic properties of classes of path ideals
我们考虑路径理想与特殊类别的偏序集，如树偏序集和周期。我们表示他们的性质是序列Cohen-Macaulay的基础偏序集。此外，由代数统计中的Luce-可分解模型产生的单项理想，可以被看作是某些偏序集的路理想。我们研究这些所谓的\emph{Luce-decomposable} monomial理想的钻石偏序集和产品的链的不变量。特别地，对于这些偏序集类，我们明确地计算了它们的克鲁尔维数、投射维数、正则性和贝蒂数.
We describe the typical homological properties of monomial ideals defined by random generating sets. We show that, under mild assumptions, random monomial ideals (RMI's) will almost always have resolutions of maximal length; that is, the projective dimension will almost always be $n$, where $n$ is the number of variables in the polynomial ring. We give a rigorous proof that Cohen-Macaulayness is a "rare" property. We characterize when an RMI is generic/strongly generic, and when it "is Scarf"---in other words, when the algebraic Scarf complex of $M\subset S=k[x_1,\ldots,x_n]$ gives a minimal f…
Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine all nonzero positions of Betti tables of $I_n$ for all large intergers $n$ from the $\mathbb Z^m$-graded Betti table of $I_m$ for some integer $m$. Our main result shows that the projective dimension and the regularity of $I_n$ eventually become linear functions on $n$, confirming a special case of conjectures posed by Le, Nagel, Nguyen and Römer.
Equations defining recursive extensions as set theoretic complete intersections
基于平面上的射影单项式曲线是完全交这一事实，给出了在射影$n$ -空间中生成无穷多条集合论上完全交的单项式曲线的一种有效的归纳方法.我们通过给出不同的无限族的例子来说明我们的主要结果。我们的证明是建设性的，并提供了一个二项式和$(n-2)$多项式显式方程的超曲面切割的曲线问题。
Proyective Cohen-Macaulay monomial curves and their affine charts
本文研究射影单项式曲线与其仿射图的坐标环 Betti 数相等的条件，提出了一种无需 Gröbner 基的组合判据作为充分条件，并据此发现了无限族满足该性质的曲线，进而改进了 Vu 关于移位曲线族 Betti 数相等最小 $j$ 值的上界。
Castelnuovo-Mumford regularity of projective monomial curves via sumsets
该研究建立了射影单项式曲线的Castelnuovo-Mumford正则度与对应整数集和集性质、齐次半群组合性质的关联，揭示了交换代数与加性数论的新联系。
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.
Minimal Resolutions of $k$-divisible and $(k-1)$-dominant Ideals
该研究引入k-可除性与d-支配性不变量，证明其与单项式理想投射维数的上下界关系，通过Lyubeznik分解给出d=k-1时的极小分解，证明Alesandroni猜想并刻画可经一次连续消去从Taylor分解得到极小自由分解的单项式理想。
Betti splitting via componentwise linear ideals
单名理想$I$允许Betti分裂$I=J+K$，如果$I$的贝蒂数可以由理想$J,K$和$J \cap K$的贝蒂数确定.给出了一个单项理想$I$，证明了$I=J+K$是$I$的Betti分裂，条件是$J$和$K$是分量线性的，推广了Francisco，Hà和Van Tuyl的结果.如果$I$具有线性分辨率，则相反也成立。我们将这个结果递归地应用于点可分解、可壳、可构造的单纯复形的Alexander对偶，并确定了射影空间中三个一般胖点的定义理想的分次贝蒂数.
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