On colon operations and special types of ideals

关于列理想生成元的理论及其在三项式近完全交集弱Lefschetz性质中的应用

The generators of a colon ideal with an application to the weak Lefschetz property for monomial almost complete intersections in three variables

本文通过研究列理想生成元，探讨了三项式近完全交集的弱Lefschetz性质，并验证了一个猜想在特定情况下的正确性。

Jordan块、微分迹与Noether商环上的余理想非包含关系

Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients

本文在Noether商环上证明幂零微分为零元素的余理想非包含定理，推广至孤立及非孤立临界轨迹，并构造反例否证积分闭包加强。

Symbolic powers of monomial ideals

Let $A = K[X_1,\ldots, X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J) = (I^n \colon J^\infty)$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f^I_J(n) = e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n \gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i = 0, \ldots, c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c \neq 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for…

理想的核的公式

A formula for the core of an ideal

一个理想的核心是它所有归约的交集。对于大类的理想I，我们明确地将核心描述为一个单一归约的幂和I的幂的冒号理想。

分次环中零因子、单位和冒号理想的齐次性

Homogeneity of zero-divisors, units and colon ideals in a graded ring

本文将Kaplansky零因子猜想和单位猜想推广至G-分次环，证明了全序群分次环中零因子、可逆元素及冒号理想的齐次性，并推广了McCoy和Armendariz的经典定理。

一类特殊单项理想的算术秩

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。

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this article, we compute the regularity of Rees algebra of binomial edge ideals of closed graphs. We obtain a lower bound for the regularity of Rees algebra of binomial edge ideals. We also study some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. We obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

关于特殊p-Borel理想

On special p-Borel ideals

为了研究一类p-Borel理想的最小分次自由分解并恢复它们的Pardue正则性公式，我们定义了约化马蹄分解和连体理想对的概念。从我们的技巧可以得出，这些理想的分次贝蒂数不依赖于基域k的特征。

Total of each of the 1-th and 2-th Betti numbers of the join-meet ideal of Special Crystal Lattice

The join-meet ideal was introduced by Takayuki Hibi in 1987. It is binomial ideals that are defined by finite lattices. We study the join-meet ideal of non-distributive finite lattices that do not always satisfy modular. In particular, we work on the case of Crystal lattice which is one of them. In this paper, we give the computational result of a total of each of the 1-th and 2-th of grade Betti numbers of the join-meet ideal of a special Crystal lattice. The important point about this result is that it is characterized by the number of combinations of special generators. Moreover, we can con…

LCM Duals of Monomial Ideals

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.

某些特殊单项理想的深度

Depth of some special monomial ideals

设$I\supsetneq J$是域上多项式代数的两个无平方根的单项理想。假设$I$由一个次数为$ d>0$的无平方根单项式和其他次数为$\geq d+1$的无平方根单项式生成。如果$I/J$的Stanley深度是$\leq d+1$，那么几乎总是$I/J$的通常深度也是$\leq d+1$。

Characterization of some special rings via linkage

Some descriptions of linked ideals in a commutative Notherian ring $R$ are provided in terms of the Associated prime ideals of $R$. Then, among other things, we make some characterization of Cohen-Macaulay, Gorenstein and regular local rings in terms of their linked ideals.

正则局部环中的完全支撑 *-单完全理想

Finitely Supported *-Simple Complete Ideals in a Regular Local Ring

设I是正则局部环（R，m）的n-支撑完全m-准素理想.李普曼的一个定理意味着我有一个唯一的因子分解作为一个特殊的 *-简单完全理想的 *-产品，其中一些因子的指数可能是负的。如果R的维数至少为3，则存在负指数，这是因为存在非特殊的n-支撑 *-单理想。我们考虑特殊 *-单完备理想的性质，如它们的Rees赋值和点基。设（R，m）是d维等特征正则局部环，m =（x_1，.，x_d）R.我们定义了R的单项二次变换，并考虑了单项理想的变换和逆变换。对于一类包含完全逆变换的单项式理想I，证明了I的生成元的最小个数完全由I的阶决定。本文给出了单项理想的 *-积的完全逆变换是因子的完全逆变换的 *-积的充要条件。这就产生了非特殊的单支 *-单项式理想的例子。证明了具有线性序基点的单支撑 *-单项式理想是特殊 *-单的.

单位区间行列式面理想的 Rees 代数

Rees Algebras of Unit Interval Determinantal Facet Ideals

本文利用 SAGBI 基技术，为单位区间行列式面理想的 Rees 代数与特殊纤维环构造 Gröbner 基，证明其纤维型、Koszul 性及正规 Cohen-Macaulay 与有理奇点性质。

Special tight closure

We prove that in normal rings the tight closure of an ideal can be computed as the sum of the ideal and a piece of the tight closure, called the special tight closure.

图中单项理想与奇洞的相伴素数

Associated primes of monomial ideals and odd holes in graphs

设$G$是一个具有边理想$I(G)$的有限单图.令$J(G)$表示$I(G)$的Alexander对偶。我们发现，所有诱导周期的奇数长度的$G$的描述编码在相关的素数的$J(G)^2$。这个结果形成了通过理想操作检测图的奇诱导圈的方法的基础，例如，交叉点、产品和冒号操作。此外，我们还得到了判定一个图是否是完美图的一个简单的代数判据。我们还展示了如何确定存在的奇洞的图的算术度的值的$J(G)^2$。

混合乘积理想的贝蒂数

Betti numbers of mixed product ideals

本文计算了一类特殊的无平方根的单项理想--混合积理想的分解的贝蒂数。此外，当这些理想是Cohen-Macaulay时，我们计算它们的型。

关于一类无平方因子单项式理想的希尔伯特深度

On the Hilbert depth of a special class of squarefree monomial ideals

研究关于无平方因子单项式理想\(I_{n,r}\)的希尔伯特深度，通过对\(S = K[x_1,\ldots,x_{n + r - 1}]\)及\(I_{n,r}\)的分析，证明相关结果，并探讨\(n = r\)时的特殊情形。

Rees algebras of sparse determinantal ideals

We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.

Graded Bourbaki ideals of graded modules

In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle an…

On the Stanley depth of a special class of Borel type ideals

We give sharp bounds for the Stanley depth of a special class of ideals of Borel type.

一类非无平方因子单项式理想幂的重数

The Multiplicity of Powers of a Class of Non-Square-free Monomial Ideals

本文研究高度为 h 的单项式理想，在其无冗余约化准素分解中高度 h 准素理想均不可约的条件下，推广已有定理，给出理想幂、特殊幂的幂，以及加权有向图边理想相应情形的重数公式。

理想的一个消去定理

A cancellation theorem for ideals

证明了Gorenstein局部环中特殊理想的消去定理。这些定理的形式是：如果KI包含在JI中，则K包含在J中。

Integral closure and Hilbert series of a special monomial ideal

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness of its integral closure. Furthermore, we compute the Hilbert series of this ideal and we show that this ideal is Freiman.

The saturation of Frobenius powers of ideals

This paper proves some special cases in which localization of tight closure holds. In particular it studies the condition LC relating to bounding the Loewy lengths of local cohomology of Frobenius iterates of quotient rings.

Computing minimal generating systems for some special toric ideals

Let $X_{P}$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_{P}$ is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining $X_{P}$ is reduced to a simple Gaussian elimination.

Multiplicity of the saturated special fiber ring of height two perfect ideals

Let $R$ be a polynomial ring and $I \subset R$ be a perfect ideal of height two minimally generated by forms of the same degree. We provide a formula for the multiplicity of the saturated special fiber ring of $I$. Interestingly, this formula is equal to an elementary symmetric polynomial in terms of the degrees of the syzygies of $I$. Applying ideas introduced in arXiv:1805.05180, we obtain the value of the j-multiplicity of $I$ and an effective method for determining the degree and birationality of rational maps defined by homogeneous generators of $I$.

Multiplicity of the saturated special fiber ring of height three Gorenstein ideals

Let $R$ be a polynomial ring over a field and $I \subset R$ be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the saturated special fiber ring of $I$. The obtained formula depends only on the number of variables of $R$, the minimal number of generators of $I$, and the degree of the syzygies of $I$. Applying results from arXiv:1805.05180, we get a formula for the $j$-multiplicity of $I$ and an effective method to study a rational map determined by a minimal set of generators of $I$.

单式猜想与阶理想

The Monomial Conjecture and Order Ideals

本文首先证明了Evans和Griffith在等特征线中提出的阶理想猜想的一个特例蕴涵了M.霍克斯特我们利用正规整环的典范模具有自由和项的某些系统，得到了这一特殊情形成立的一个充要条件。我们还证明了这种观察的一些特殊情况。

一致增超图的包含理想

Inclusion Ideals Associated to Uniformly Increasing Hypergraphs

本文引入了一类特殊的非一致超图H（X; E; d）的单项式理想I（H），即一致增超图。这些理想被称为包含理想。本文讨论了这类包含理想的一些代数性质。特别地，我们给出了包含理想的特殊对偶理想I^[*]（H）的Castlenouvo-Mumford正则性的一个上界.

对角理想的Rees代数

Rees Algebras of Diagonal Ideals

理想的对称代数到Rees代数存在一个自然满射。当这个满射是同构时，我们说理想是线性型的。给定域上的两个行列式环，我们考虑乘法映射的核-对角理想。证明了对角理想是线性型的，并在某些特殊情况下恢复了Rees代数的定义理想。对角理想的特殊纤维环是并簇的齐次坐标环。

广义安德拉斯法伊图和边理想特殊贝蒂图

Generalized Andrásfai graphs and special Betti diagrams of edge ideals

研究广义安德拉斯法伊图的边理想特性，发现其贝蒂图具有特殊形状，扩展了完整图和完整双图的正则度和射影维数结果。

对偶紧闭理论

A dual to tight closure theory

引入了特征为$p$的$F$ -有限环上模的运算.我们称之为\emph{tight interior}。虽然它更普遍地存在，但在某些情况下，这种操作等价于紧闭的Matlis对偶。此外，戒指本身的内部只是大测试的理想。我们直接证明，没有呼吁紧密封闭，结果类似于持久性，结肠捕获，和工作模最小素数，我们开始开发一个理论双幻影同源。利用我们的对偶概念的持久性，我们获得了新的和有趣的变换规则的紧内部，因此特别是测试理想，补充了最近的论文的主要结果的第二作者和K。塔克利用幻影同调理论，我们证明了Ext.我们还比较了我们的理论与M。Blickle的Cartier模的概念，并在此过程中，我们证明了Blickle的测试子模的新的存在性结果。最后，我们将所发展的理论应用于非正规环中的测试理想的研究，证明了有限测试理想在某些情况下与大测试理想重合。

Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas

We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that …

边理想幂的正则性

The Regularity of Powers of Edge Ideals

本文证明了任意图的边理想的幂的极小单项生成元集上的一个特殊序的存在性。利用这个序，我们找到了补图没有诱导四圈的图的边理想的幂的正则性的新的上界。