Questions on Local Cohomology and Tight Closure Theory

关于局部上同调和紧闭理论的问题

• 批准号: 1500613
• 负责人: Anurag Singh
• 金额: $ 22万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500613&HistoricalAwards=false
• 关键词: Questions; Local; Cohomology; Tight; Closure

This project is concerned with several questions in commutative algebra. This is a field that studies solution sets of polynomial equations. Understanding solution sets of polynomial equations is of fundamental importance in many sciences, in engineering, and in other disciplines as well. Most of the questions that will be investigated deal with the nature of the solution sets: some are questions that have been unresolved for a number of years, but for which recent advances provide the likelihood of a solution; others arise naturally from new developments in the area. A number of projects are connected with local cohomology theory: this theory often provides the best answers to basic questions such as the least number of polynomials needed to define a solution set.The projects on local cohomology theory include algorithmic aspects, as well as structural properties such as support and injective dimension. There is a special focus on local cohomology modules of polynomial rings and hypersurfaces 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 work of the PI and collaborators. The project will also investigate local cohomology over the integers. The research will further develop the connections of local cohomology with prime characteristic numerical invariants such as the F-pure threshold, and will study the composition series of local cohomology modules over rings of differential operators.

该项目涉及交换代数中的几个问题。 这是研究多项式方程组解集的领域。 理解多项式方程的解集在许多科学、工程和其他学科中都具有根本性的重要性。 将要研究的大多数问题都涉及解决方案集的性质：有些问题多年来一直未得到解决，但最近的进展提供了解决方案的可能性；其他的则自然地从该领域的新发展中产生。 许多项目与局部上同调理论相关：该理论通常为基本问题提供最佳答案，例如定义解集所需的最少多项式数量。局部上同调理论的项目包括算法方面，以及结构特性，例如支持度和单射维数。 特别关注整数上多项式环和超曲面的局部上同调模：这源于整数到任何环都存在规范同态的事实，这使得整数上的局部上同调模在某种意义上是通用的；这种观点在 PI 和合作者最近的工作中被证明是有用的。该项目还将研究整数的局部上同调。 研究将进一步发展局部上同调与F-纯阈值等素特征数值不变量的联系，并研究微分算子环上局部上同调模的组成级数。

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

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