Nearly Gorenstein projective monomial curves of small codimension
Sora Miyashita
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。
本文刻画了余维数为2和3的近Gorenstein射影单项曲线。
In this paper, we characterize nearly Gorenstein projective monomial curves of codimension 2 and 3.
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We extend some results on almost Gorenstein affine monomial curves to the nearly Gorenstein case. In particular, we prove that the Cohen-Macaulay type of a nearly Gorenstein monomial curve in $\mathbb{A}^4$ is at most $3$, answering a question of Stamate in this particular case. Moreover, we prove that, if $\mathcal C$ is a nearly Gorenstein affine monomial curve which is not Gorenstein and $n_1, \dots, n_ν$ are the minimal generators of the associated numerical semigroup, the elements of $\{n_1, \dots, \widehat{n_i}, \dots, n_ν\}$ are relatively coprime for every $i$.
We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve $\mathcal C \subseteq \mathbb{A}^4$ is at most $3$, and make some considerations on the general case.
We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel.
We study the minimal free resolution of the tangent cone of Gorenstein monomial curves in affine 4-space. We give the explicit minimal free resolution of the tangent cone of non-complete intersection Gorenstein monomial curve whose tangent cone has five minimal generators and show that the possible Betti sequences are $(1,5,6,2)$ and $(1,5,5,1)$. Also, we compute the Hilbert function of the tangent cone of these families as a result.
A note on Gorenstein monomial curves
设$k$为任意字段。本文证明了：若一个互质正整数序列${\bf a}=(a_1,a_2,a_3,a_4)$在${\mathbb A}_k^4$中定义了一条Gorenstein非完全交单项曲线${\mathcal C}({\bf a})$，则存在两个向量${\bf u}$和${\bf v}$，使得对几乎所有$t\geq 0$，${\mathcal C}({\bf a}+t{\bf u})$和${\mathcal C}({\bf a}+t{\bf v})$也是Gorenstein非完全交仿射单项曲线.
Structure theorems for Gorenstein ideals of codimension four with small number of generators
本文用表示论方法研究余维数为四的Gorenstein理想的极小自由消解，引入高阶结构映射族，并证明由六个元素极小生成的该类理想必为余维数为三的Gorenstein理想的超平面截面，同时提出七、八个生成元情形的猜想。
Koszul homology of codimension 3 Gorenstein ideals
本文计算了余维为3的Gorenstein理想的Koszul同调.我们发现过滤的Koszul同调的模块与纯自由分解，并完全描述这些决议。我们还考虑Huneke-Ulrich偏差2理想。
Log-concave Gorenstein sequences
该研究探讨不同余维数下相关序列的对数凹性，证明3维余Artinian Gorenstein序列对数凹、存在4维余非对数凹该类序列，且2维余level序列及任意余维compressed level希尔伯特函数均对数凹。
Stanley's theorem on codimension 3 Gorenstein $h$-vectors
本文给出了R的下列著名定理的初等证明。斯坦利：余维为3的Gorenstein代数的$h$ -向量是SI-序列，即是对称的，并且它们的前半部分的第一差是$O$ -序列。
An improved Multiplicity Conjecture for codimension three Gorenstein algebras
重数猜想是Cohen-Macaulay标准分次代数在其最小自由分解下的重数（或度）与某些极值分次贝蒂数之间的一个深层次问题.在余维为3的水平代数的情况下，Zanello提出了一个更强的猜想。对于余维三次Gorenstein代数的情况，我们证明了这个猜想。
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determine…
Vanishing of cohomology over Gorenstein rings of small codimension
证明了若M，N是余维数至多为4的Gorenstein局部环R上的有限模，则Ext^n_R（M，N）对n个\gg 0的消失等价于Ext^n_R（N，M）对n个\gg 0的消失.此外，如果$R$的完备化没有嵌入变形，则当且仅当M或N具有有限的投射维数时，这种消失才会发生。
A characterization of Gorenstein Hilbert functions in codimension four with small initial degree
本文的主要目标是刻画所有（artinian）余维4 Gorenstein代数的Hilbert函数，这些代数至少有两个独立的四度关系。这包括所有余维4的Gorenstein代数，其初始关系的度至多为3。我们的结果表明，这些Hilbert函数正是以（1，4，h_2，h_3，.）开始的{ \em SI-序列}，其中$h_4 \leq 33$。特别地，这些希尔伯特函数都是单峰的。我们还建立了一个更一般的单峰结果，它依赖于Hilbert函数的值不是太大，但与初始度无关。
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
We prove that the Cohen-Macaulay type of an almost Gorenstein monomial curve $\mathcal{C} \subseteq \mathbb{A}^5$ is bounded.
The aim of the article is to study the Betti numbers of the tangent cone of Gorenstein monomial curves in affine 4-space. If $C_S$ is a non-complete intersection Gorenstein monomial curve whose tangent cone is Cohen-Macaulay, we show that the possible Betti sequences are (1,5,5,1), (1,5,6,2) and (1,6,8,3).
The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.
Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves
本文研究 4 维空间中 Gorenstein 非完全交单项式曲线的切锥 Cohen-Macaulay 性，利用定义理想的极小生成元集给出判定准则，并刻画了相应理想极小生成元个数的可能取值。
On Unimodality of Hilbert Functions of Gorenstein Artin Algebras of Embedding Dimension Four
证明了余维4分次Gorenstein-Artin代数R/I的Hilbert函数是单峰的，只要I有一个小于5次的极小生成元.是否所有余维为4的Gorenstein h-向量都是单峰的，这是一个悬而未决的问题。本文证明了所有以（1，4，10，20，h_4，.）开始的Artin余维4 Gorenstein代数的Hilbert函数，其中h_4 \leq 34是单峰的。结合已有的结果，我们得到所有Gorenstein h-向量（1，h_1，h_2，h_3，h_4，.）如果h_4 \leq 3.
Cohen-Macaulay Monomial Ideals of Codimension 2
给出了余维为2的Cohen Macaulay单项理想的一个结构定理，并描述了这类理想的所有可能的关系矩阵。在理想有线性分解的情况下，关系矩阵可以用一个连通弦图的生成树来表示，并且该图的每对极大团至多有一个公共点.
Let $C$ be a Gorenstein non complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the minimal number of generators of the tangent cone of $C$. Special attention will be paid to the case where $C$ has Cohen-Macaulay tangent cone at the origin.
In this paper, we consider monomial curves in ${\mathbb A}_k^3$ parameterized by $t \rightarrow (t^{2q +1}, t^{2q +1 + m}, t^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. The symbolic blowup algebras of these monomial curves is Gorenstein (\cite{goto-nis-shim}, \cite{goto-nis-shim-2}). We give a simple proof for the the Gorenstein property for the symbolic blowup algebras of these curves.
We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. We apply our technique to various classes of algebras, including monomial almost complete intersections and Gorenstein algebras. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property. We also study the reverse of the splitting operation -- a gluing operation -- which gives a way to construct monomial algebras with the Lefschetz properties.
On posets, monomial ideals, Gorenstein ideals and their combinatorics
本文从组合角度比较单项式理想的 socle 元素与 Macaulay 逆系统，建立相关偏序集的闭包性质，推导 Γ-分次环的代数命题及其组合推论，并对零维 Gorenstein 理想进行直接计算，揭示多项式环与局部环之间的关键差异。
Bounds and asymptotic minimal growth for Gorenstein Hilbert functions
我们确定新的边界上的条目Gorenstein希尔伯特函数，无论是在任何固定的余维和渐近。我们的第一个主要定理是Gorenstein $h$ -向量的度$i+1$入口的下界，根据它在度$i$中的入口。这个结果进行有趣的应用单峰：事实上，一个重要的后果是，给定$r$和$i$，所有Gorenstein $h$ -向量的余维$r$和柱脚度$e\geq e_0=e_0(r,i)$（这个函数被显式计算）是单峰的程度$i+1$。这立即给出了一个新的证明定理的斯坦利，所有Gorenstein $h$ -向量余维3是单峰的。我们的第二个主要定理是一个渐近公式的最小值，第$i$项的Gorenstein $h$ -向量可以假设，在余维，$r$，和socle度，$e$。这个定理广泛地推广了我们最近的一个结果，在那里我们证明了斯坦利的一个猜想，预测在特定情况下$e=4$和$i=2$的渐近值，以及Kleinschlaughter的结果，其中涉及的对数渐近行为的程度$i= \lfloor \frac{e}{2} \rfloor $。
We study Gorenstein ideals of codimension $4$ derived from generic doublings of almost complete intersection perfect ideals of codimension $3$. We also investigate spinor coordinates of such Gorenstein ideals with $8$ and $9$ generators. For an ideal $J$ of commutative ring $R$, the $R/J$ module $J/J^2$ is called conormal module and $R/J$-dual of $J/J^2$ is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension $3$.
Jordan degree type for codimension three Gorenstein algebras of small Sperner number
该研究将余维数2分次代数的Jordan次数型研究拓展至小Sperner数的余维数3分次Artinian Gorenstein代数，完全确定了s=3,4,5时对应Gorenstein序列的所有可能JDT，并划定了s=6时的潜在JDT范围。
On the Betti numbers of some Gorenstein ideals
设$R$是域上的多项式环，$I$是余维为$g\ge3$且初度为$p\ge2$的齐次Gorenstein理想.证明了$I$的在度$p$中的极小生成元$ν(I_p)$的个数由$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$上有界，$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$是余维$g$和初度$p$的极值Gorenstein代数的定义理想的极小生成元个数.此外，如果$ν(I_p)=ν_0$，则$I$本身是极值的。
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, f…
In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as the Small Cohen-Macaulay Conjecture reduce to the excellent unique factorization domains. Some other reductions of the Monomial Conjecture are also proved. We, moreover, show that certain almost complete intersections adhere the Monomial Conjecture.
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
Monomial ideals and the Gorenstein liaison class of a complete intersection
众所周知，对于余维为2的${\mathbb P}^{n}$中的子方案$V$，条件（1）$V$是ACM，以及（2）$V$是“licci”（即$V$处于完整交集的联络类中）是等价的。在较高的余维中，（2）蕴涵（1），但反之则不成立。对于Gorenstein联络，（2）的类似物是：$(2')$ $V$是“glicci”（即$V$在完全交集的Gorenstein联络类中）。$(2')$仍然意味着（1），但现在反过来是一个重要的开放问题。以前，作者描述了一种机制，提高单项式理想减少工会的线性品种。我们在这里表明，如果单项理想是阿丁然后相应的工会是glicci。因此，任何出现在ACM方案中的希尔伯特函数实际上都出现在glicci方案中。如果我们限制为完整的交叉连接，则情况并非如此。我们还证明了任何科恩-麦考利-博雷尔固定单名理想都是glicci。因此，射影空间的所有ACM子概型在平坦变形下都是glicci.
On the weak Lefschetz property of Artinian Gorenstein algebras of codimension three in arbitrary characteristic
本文将特征零代数闭域上三维余维Artinian Gorenstein代数的弱Lefschetz性质结论推广到任意特征的无限域，同时证明了相关的公因子结论。
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.
Gorenstein in codimension 4 - the general structure theory
我描述了余维4 Gorenstein理想的投射分解，旨在推广Buchsbaum和Eisenbud在余维3的著名结果。主要结果是一个结构定理，指出理想是由它的（k+1）× 2k矩阵的第一合合，看作是一个态射从周围的正则空间的Spin-Hom品种SpH_k在Mat（k+1，2k）。这是一个概括性的结果，包含了问题的一些理论方面，但是，就目前而言，离易于处理的应用还有一段距离。
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.
L. Szpiro's conjecture on Gorenstein algebras in codimension 2
余维为2的Gorenstein A-代数R是完全有限A-代数，使得R=Ext^2（R，A）作为R-模成立，A是Cohen-Macaulay局部环，dim（A）-dim_A（R）=2.证明了这类代数的一个结构定理，改进了M.格拉西特别注意的问题是如何环结构的R编码在其希尔伯特决议。本文证明了当对由R在A上的表示矩阵导出的行列式理想施加弱深度条件时，R自动成为环。讨论了Gorenstein代数与M.格拉西澄清。P^4中的正则曲面的适用性问题在这些研究中起到了指导作用。
Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Oriented Artinian Gorenstein Algebras in Codimension Two
该研究引入$i$-洛伦兹多项式，证明其为余维2有向阿廷戈伦斯坦代数的Macaulay对偶生成元，建立其与全非负Toeplitz矩阵的一一对应，推导相关推论并给出新的Toeplitz矩阵惠特尼型定理。
New families of Artinian Gorenstein algebras with the weak Lefschetz property
本文构造了任意余维数且具有二项式 Macaulay 对偶生成元的 Artinian Gorenstein 代数新族，并证明其满足弱或强 Lefschetz 性质，是相关姊妹篇的补充。
Artinian Gorenstein algebras with binomial Macaulay dual generator
本文系统研究具有二项式Macaulay对偶生成元的Artinian Gorenstein代数，在余维数3证明其满足强Lefschetz性质并给出完全交刻画，同时为任意余维数建立最优的弱Lefschetz性质充分条件。
Canonical surfaces in P^4 and Gorenstein algebras in codimension 2
本文研究了p_g=5，q=0的一般型极小曲面（所谓P^4中的典型曲面），其中1-典型映射是P^4中曲面上的双有理态射，并将其看作P^4的齐次坐标环上的余维为2的Gorenstein代数，通过上述曲面的典型环的Hilbert分解的结构定理得到了上述曲面。我讨论如何环结构这样的代数编码在其决议。其中，我展示了如何应用这种方法来分析具有p_g=5，q=0，K^2=11的正则曲面的模空间，从而恢复了D.罗斯伯格用不同的技术。
On the initial ideal of a generic artinian Gorenstein algebra
本文研究一般阿廷戈伦斯坦代数的初始理想，证明一般形式零化理想的初始理想由各次数最大单项式生成，并确定一类强稳定单项理想的总Betti数与相应初始理想一致。
On the Weak Lefschetz Property for Artinian Gorenstein algebras of codimension three
研究了任意余维三阶Artin Gorenstein代数是否具有弱Lefschetz性质的问题。我们减少这个问题，以检查它是否持有所有压缩Gorenstein代数的奇数socle度。在第一个开放的情形，即Hilbert函数（1，3，6，6，3，1）中，我们通过将问题转化为研究从$\mathbb P^2$到$\mathbb P^3$的某些态射的几何方面，以及$\mathbb P^2$中的Hesse配置，给出了每个特征的完整答案.
The strong Lefschetz property in codimension two
如果$K$是特征为零的域或者是特征大于商的正则性的无限域，则$K[x,y]$的每一个Artin商都具有强Lefschetz性质.我们改进了这个界限的情况下，单项理想。当两个边界都是尖锐的时，我们使用这一点进行分类。证明了$K[x,y]$中的单项理想的Artin商具有强Lefschetz性质，而与域的性质无关，只要理想是lexsegment.作为结果，我们描述了一个家庭的非单项完全交叉，总是有很强的莱夫谢茨财产。
Roller Coaster Gorenstein algebras and Koszul algebras failing the weak Lefschetz property
受图论过山车定理启发，通过须状图定义的二次单项式代数的Nagata理想化构造，证明无约束希尔伯特级数的过山车Gorenstein代数存在，发现大量不满足弱Lefschetz性质的G-二次Gorenstein代数。
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
本文证明了基座度数为d且Sperner数至多为d+1的artinian Gorenstein代数均满足弱Lefschetz性质，且与余维数无关，并指出该界限在一般情况下是严格的。
Gorenstein Liaison and ACM Sheaves
研究了算术Gorenstein概型X的余维两个子概型的Gorenstein联络.我们的主要结果是两个这样的子方案在同一Gorenstein联络类，在X上的ACM层的范畴。作为结果，我们得到了一个标准，X有的性质，每个余维2算术Cohen-Macaulay子计划是在Gorenstein联络类的一个完整的交集。利用这些工具，我们证明了$\mathbb{P}^n$的每一个算术Gorenstein子概型都在完全交的Gorenstein联络类中，并且我们能够刻画$\mathbb{P}^4$中非奇异二次曲面三重上曲线的Gorenstein联络类.
Stanley decompositions and partitionable simplicial complexes
我们研究了Stanley分解，证明了Stanley关于Stanley分解的猜想蕴含着他关于可分Cohen-Macaulay单纯复形的猜想.我们还证明了这些定理的所有Cohen-Macaulay单项理想的余维2和所有Gorenstein单项理想的余维3。
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
Liaison addition and the structure of a Gorenstein liaison class
本文研究了算术Gorenstein射影概形余维两个子概型的联络加法的概念。我们展示了它如何涉及联络和biliaison类的子计划，并用它来调查Gorenstein联络等价类的结构，扩展已知的理论，为完整的交叉联络余维两个子计划。特别地，我们证明了在射影四维空间中的非奇异三重二次曲面上，每一条非licci ACM曲线都可以由一条直线通过与直线和CI-双直线的连续联络加法得到.
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four
本文针对余维数4、正则度不超过6的Artinian Gorenstein环，研究其极小自由分解与弱/强Lefschetz性质及Jordan型之间的关系。
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of th…
Stanley's nonunimodal Gorenstein h-vector is optimal
本文对分次Artin Gorenstein代数的所有可能的$h$ -向量进行了基点度为4且余维为$\leq 17$和基点度为5且余维为$\leq 25$的分类.我们得到的结果，允许存在一个非单峰Gorenstein $h$ -向量的变量的最少数量是13为4度的基座，和17为5度的基座。特别是，最小的非单峰Gorenstein $h$ -向量是$(1,13,12,13,1)$，它是由Stanley在他1978年关于水平代数的开创性论文中构造的。这解决了这个领域长期存在的一个问题。我们所有的结果都是特征自由的。
Gluing and Hilbert functions of monomial curves
本文利用半群的胶合技巧，给出了无穷多个具有非减Hilbert函数的一维局部环族.更重要的是，这些是局部环，其相关的分次环不一定是Cohen-Macaulay。在这个意义上，我们给出了构造一族与单项曲线相关的Gorenstein局部环的有效方法，支持了Rossi猜想：每个Gorenstein局部环都有非减Hilbert函数.
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to al…
On monomial curves obtained by gluing
利用胶合的概念，研究了与仿射单项曲线相关联的切锥的算术性质。特别是，我们刻画了Cohen-Macaulay和Gorenstein性质的切锥的一些家庭的单式曲线得到的胶合。此外，我们提供了新的单式曲线族的非减希尔伯特函数。
On nearly Gorenstein affine semigroups
本文通过描述单纯仿射半群环的典范模与迹理想，用半群算术性质刻画了近Gorenstein性，并给出近Gorenstein情形下Cohen-Macaulay型的界，证明余维数至多三时该型至多为三且界是精确的。
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