Determinantal Rings, Local Cohomology, and Tight Closure

该项目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都会自然出现，而交换代数继续与多个领域发展出令人着迷的相互作用，成为科学和工程中越来越有价值的工具。该项目的重点是与局部上同调、紧闭理论和经典不变量环相关的交换代数问题；所有这些问题都是从最近的事态发展中自然产生的。局部上同调通常为基本问题（例如定义解集所需的最少多项式方程数量）提供最佳答案。这个方向的项目包括算法方面以及结构特性；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。这种观点在最近与 Lyubeznik 和 Walther 的合作中被证明是有用的。与此同时，与 Bhatt、Blickle、Lyubeznik 和 Zhu 的合作开发了研究整数局部上同调的新技术；建议将这些新技术扩展到算法中。与紧闭理论相关的项目包括调查 Hankel 行列式环的奇点，这是最近与 Conca、Mostafazadehfard 和 Varbaro 联合​​工作提出的问题，以及这些环是否作为线性还原群作用的不变环出现。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.