Specializations of Ferrers ideals

Ferrers图的覆盖理想幂的投射维数

Projective dimension of powers of cover ideal of Ferrers graphs

本文通过分析Ferrers图对应的覆盖理想J_λ的幂，证明了其投射维数等于min{t, d(λ)}+1，其中d(λ)是λ的不同部分数。

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 …

Specialization of Integral Closure of Ideals by General Elements

In this paper, we prove a result similar to results of Itoh and Hong-Ulrich, proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form.

Betti numbers of monomial ideals and shifted skew shapes

We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certain classes of ideals that generalize (in two different directions) the edge ideals of threshold graphs and Ferrers graphs. In the process, we produce particularly simple cellular linear resolutions for strongly stable and squ…

Specialization and Integral Closure

We prove that the integral closedness of any ideal of height at least two is compatible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an element is integral over a module if it is integral modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal.

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…

Enumeration of concave integer partitions

An integer partition λof n corresponds, via its Ferrers diagram, to an artinian monomial ideal I of colength n in the polynomial ring on two variables. If the partition λcorresponds to an integrally closed ideal we call λconcave. We study generating functions for the number of concave partitions, unrestricted or with at most r parts.

Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings

The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by …

Linear strands of edge ideals of multipartite uniform clutters

We construct the first linear strand of the minimal free resolutions of edge ideals of $d$-partite $d$-uniform clutters. We show that the first linear strand is supported on a relative simplicial complex. In the case that the edge ideals of such clutters have linear resolutions, we give an explicit and surprisingly simple description of their minimal free resolutions, generalizing the known resolutions for edge ideals of Ferrers graphs and hypergraphs and co-letterplace ideals. As an application, we show that the Lyubeznik numbers that appear on the last column of the Lyubeznik table of the co…

通过Apéry特殊化得到数值半群代数的极小自由分解

Minimal free resolutions of numerical semigroup algebras via Apéry specialization

本文基于Kunz关于数值半群由多面锥整点参数化的结论，构造了toric理想的自由分解，其结构对锥固定面相对内部参数化的半群一致，且在最大嵌入维数时达到极小性。

Bounding the degrees of the defining equations of Rees rings for certain determinantal and Pfaffian ideals

We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of other ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the Rees rings of a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, …

加权 Borel 生成元

Weighted Borel Generators

本文提出强稳定理想的专门化，通过权重向量限制生成元子集定义 $w$-稳定理想及加权 Borel 生成元，并开发 Macaulay2 软件包支持相关计算。

Minimal and cellular free resolutions over polynomial OI-algebras

Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely minimal and width-wise minimal free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say $i$, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algeb…

余维数2的Cohen-Macaulay代数的典范迹

The canonical trace of Cohen-Macaulay algebras of codimension 2

本文证明了Herzog关于余维数2的Cohen-Macaulay代数典范迹可由极大子式泛型情形特殊化得到的猜想，并分类了高度为2的近Gorenstein单项式理想。