Determinantal Rings, Local Cohomology, and Tight Closure

行列式环、局部上同调和紧闭

• 批准号: 1801285
• 负责人: Anurag Singh
• 金额: $ 25.5万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1801285&HistoricalAwards=false
• 关键词: Determinantal; Rings; Local; Cohomology; Tight

The project is concerned with several questions in commutative algebra: this is a field that studies solution sets of polynomial equations by means of studying polynomial functions on the solution sets. Polynomial equations arise naturally in a number of situations, and commutative algebra continues to develop a fascinating interaction with several fields, becoming an increasingly valuable tool in science and engineering. The focus of the project is on questions in commutative algebra relating to local cohomology, tight closure theory, and classical rings of invariants; all of these questions arise quite naturally from recent developments. Local cohomology often provides the best answers to fundamental questions such as the least number of polynomial equations needed to define a solution set. Projects in this direction include algorithmic aspects as well as structural properties; there is a special focus on local cohomology modules of polynomial rings over the integers: this stems from the fact that there is a canonical homomorphism from the integers to any ring, and this makes local cohomology modules over the integers, in a sense, universal. This viewpoint has proved useful in recent joint work with Lyubeznik and Walther. At the same time, new techniques for investigating local cohomology over the integers have been developed in joint work with Bhatt, Blickle, Lyubeznik, and Zhang; it is proposed to extend these new techniques to an algorithm. Projects related to tight closure theory include investigating the singularities of Hankel determinantal rings, a question coming from recent joint work with Conca, Mostafazadehfard, and Varbaro, and whether these rings arise as invariant rings for actions of linearly reductive groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

本课题涉及交换代数中的几个问题：这是一个通过研究解集上的多项式函数来研究多项式方程解集的领域。多项式方程在许多情况下自然出现，交换代数继续与几个领域发展出迷人的相互作用，成为科学和工程中越来越有价值的工具。该项目的重点是交换代数中有关局部上同调、紧闭理论和经典不变量环的问题；所有这些问题都很自然地出现在最近的事态发展中。局部上同调通常为基本问题提供最佳答案，例如定义解集所需的多项式方程的最少数量。这个方向的项目包括算法方面以及结构特性；我们特别关注整数上多项式环的局部上同调模：这源于整数到任何环都有一个正则同态，这使得整数上的局部上同调模，在某种意义上是全称的。这一观点在最近与吕别兹尼克和瓦尔特的联合工作中被证明是有用的。与此同时，与Bhatt， Blickle， Lyubeznik和Zhang共同研究了整数上的局部上同性的新技术；提出将这些新技术扩展为一个算法。与紧密闭包理论相关的项目包括调查Hankel确定性环的奇异性，这是最近与Conca， mostafazadehhard和Varbaro共同研究的问题，以及这些环是否会作为线性约化群作用的不变环出现。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Stabilization of the cohomology of thickenings

• DOI: 10.1353/ajm.2019.0013
• 发表时间: 2016-05
• 期刊: American Journal of Mathematics
• 影响因子: 1.7
• 作者: B. Bhatt;Manuel Blickle;G. Lyubeznik;Anurag Singh;Wenliang Zhang

Koszul and local cohomology, and a question of Dutta

Koszul 和局部上同调，以及 Dutta 问题

• DOI: 10.1007/s00209-020-02619-0
• 发表时间: 2021
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Ma, Linquan;Singh, Anurag K.;Walther, Uli
• 通讯作者: Walther, Uli

On a conjecture of Lynch

关于林奇的猜想

• DOI: 10.1080/00927872.2020.1722821
• 发表时间: 2020
• 期刊: Communications in Algebra
• 影响因子: 0.7
• 作者: Singh, Anurag K.;Walther, Uli
• 通讯作者: Walther, Uli

An asymptotic vanishing theorem for the cohomology of thickenings

加厚上同调的渐近消失定理

• 发表时间: 2021
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Bhatt, Bhargav;Blickle, Manuel;Lyubeznik, Gennady;Singh, Anurag K.;Zhang, Wenliang
• 通讯作者: Zhang, Wenliang

Homogeneous prime elements in normal two-dimensional graded rings

普通二维渐变环中的齐次素数元素

• DOI: 10.1016/j.jalgebra.2018.07.012
• 发表时间: 2018
• 期刊: Journal of Algebra
• 影响因子: 0.9
• 作者: Singh, Anurag K.;Takahashi, Ryo;Watanabe, Kei-ichi
• 通讯作者: Watanabe, Kei-ichi