Tight Closure, Local Cohomology, and Related Questions

所提出的研究问题源于交换代数中长期存在的问题和猜想。这些问题与Hochster和Huneke的紧闭理论、同调猜想和局部上同调理论有关。PI将寻求一种方法来解决Hochster的单项式猜想，它位于这三个主题的交叉点。这个猜想对于不包含场的环是无法解决的，例如那些出现在数论中的环。该方法涉及用任意低值的元素湮灭阻塞局部上同模的元素。这个想法在海特曼的工作中得到了强有力的证明，他解决了维数为3的环的单项式猜想。获得这种局部上同湮灭子的描述是一个庞大的计划，建议的研究将集中在一些具体的初始情况上。在与Uli Walther的合作中，PI将研究Lyubeznik关于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，现在在各种情况下都知道了这一点，但对于整数上的多项式环仍然没有解决。交换代数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，而交换代数的主要研究对象是这些解集上的函数。本文研究的大部分问题是关于方程组解的存在性和解集的性质的问题。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、密码学和其他战略应用中越来越有价值的工具。

The proposed research problems stem from long-standing questions and conjectures in commutative algebra. These are related to the tight closure theory of Hochster and Huneke, to the homological conjectures, and to the theory of local cohomology. The PI will pursue an approach to Hochster's monomial conjecture which lies at the intersection of these three topics. This conjecture is unresolved for rings which do not contain a field, such as those which arise in number theory. The proposed approach involves annihilating the elements of obstruction local cohomology modules by elements of arbitrarily low valuation. This idea has proved remarkably strong in the work of Heitmann, where he settled the monomial conjecture for rings of dimension up to three. Obtaining a description of such annihilators of local cohomology is a vast program, and the proposed research will focus on some concrete initial cases. In joint work with Uli Walther, the PI will work on Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers.Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are functions on these solution sets. Most of the questions which will be investigated in the proposed research are questions about the existence of solutions for families of equations, and 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 applications of strategic interest.