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.

该项目涉及交换代数中的几个问题：这是一个领域，该领域通过研究溶液集的多项式函数来研究多项式方程的解决方案集。多项式方程在许多情况下自然出现，而交换代数继续与几个领域建立迷人的互动，成为科学和工程学中越来越有价值的工具。该项目的重点是与当地同胞，紧密闭合理论和不变的经典戒指有关的交换代数的问题。所有这些问题都自然而然地来自最近的发展。当地的同时学通常可以为基本问题提供最佳答案，例如定义解决方案集所需的最少的多项式方程。朝这个方向的项目包括算法方面和结构特性；特殊的重点是整数上多项式环的局部共同体学模块：这源于以下事实：以下事实：从整数到任何环都有规范的同态，这使整数上的局部共同体学模块在一个通用中。在与Lyubeznik和Walther最近的联合合作中，该观点已被证明很有用。同时，在与Bhatt，Blickle，Lyubeznik和Zhang的联合合作中开发了用于研究整数的本地共同体学的新技术。建议将这些新技术扩展到算法。与紧密闭合理论相关的项目包括调查汉克尔决定性戒指的奇点，来自最近与conca，mostafazadehfard和varbaro的联合合作提出的问题，以及这些戒指是否作为线性还原群体的行动而出现的戒指。这些奖项反映了NSF的法定任务和构成的范围。

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