Questions on Local Cohomology and Tight Closure Theory

这个项目涉及交换代数中的几个问题。 这是一个研究多项式方程解集的领域。 了解多项式方程的解集在许多科学、工程和其他学科中具有根本的重要性。 大多数的问题，将调查处理的性质的解决方案集：一些问题，已经解决了数年，但最近的进展提供了一个解决方案的可能性;其他自然产生的新的发展领域。 许多项目都与局部上同调理论有关：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，以及结构性质，如支撑和内射维数。 有一个特别关注的局部上同调模的多项式环和超曲面的整数：这源于这样一个事实，即有一个典型的同态从整数到任何环，这使得局部上同调模的整数，在某种意义上，普遍的;这一观点已被证明是有用的，在最近的工作PI和合作者。该项目还将研究整数上的局部上同调。 该研究将进一步发展局部上同调与素特征数值不变量如F-纯阈值的联系，并将研究微分算子环上局部上同调模的合成序列。

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.