Local cohomology, tight closure, and related questions

PI将研究交换代数中的几个问题，这些问题以局部上同调理论、同调猜想和紧闭理论为中心。PI将继续研究正则环的局部上同模有有限多个相关素数理想的猜想。由于Huneke-Sharp和Lyubeznik的工作，这在各种情况下是已知的，但对于整数上的多项式环仍然没有解决。最近在与柳别兹尼克和瓦尔特的联合工作中解决了一些关键案件；这些反过来又引出了局部上同调模的新的消失定理。对于不包含场的环，如数论中出现的环，Hochster的单项式猜想仍然没有解决；PI将通过局部上同调来寻求解决这个问题的方法。在紧闭中提出的工作包括研究弱$F$-正则性——环的所有理想都是紧闭的性质——在局域化下是否保留；这将通过分裂环的概念进行攻击。这个项目是关于交换代数的问题。这是一个与代数几何密切相关的领域：代数几何关注多项式方程解集的几何，而交换代数的观点是研究解集上的多项式函数环。研究的大多数问题都是关于解集性质的问题。交换代数继续与数学的几个分支发展着迷人的相互作用，并且正在成为工程、编码理论、密码学和其他战略兴趣领域中越来越有价值的工具。

The PI will work on several problems in commutative algebra that are centered around local cohomology theory, the homological conjectures, and the theory of tight closure. The PI will continue work on the conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers. Some critical cases have been settled recently in joint work with Lyubeznik and Walther; these in turn, lead to new vanishing theorems for local cohomology modules. Hochster's monomial conjecture remains unresolved for rings that do not contain a field, such as those arising in number theory; the PI will pursue an approach to this via local cohomology. The proposed work in tight closure includes investigating the question whether weak $F$-regularity---the property that all ideals of a ring are tightly closed---is preserved under localization; this will be attacked via the notion of splinter rings. This project is concerned with questions in commutative algebra. This is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solution sets of polynomial equations, the point of view in commutative algebra is to study the ring of polynomial functions on a solution set. Most of the questions that will be investigated are questions about the nature of the solution sets. Commutative algebra continues to develop a fascinating interaction with several branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other areas of strategic interest.