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假设。这些都涉及到紧封闭理论的Hochster和Huneke，以同调代数，并以理论的地方上同调。PI将追求一种方法，以霍希斯特的单项猜想，在于这三个主题的交叉点。这个猜想是解决环不包含一个领域，如那些出现在数论。所提出的方法涉及零化的障碍局部上同调模的元素的任意低的价值。这一想法已被证明是非常强大的工作海特曼，他解决了单项猜想环的维数高达3。获得这样的零化子的局部上同调的描述是一个庞大的计划，和拟议的研究将集中在一些具体的初始情况。在与乌利·瓦尔特的联合工作中，PI将致力于Lyubeznik的猜想，即正则环的局部上同调模有1000个相关的素理想。这是现在已知的各种情况下，由于工作的Huneke夏普和Lyubeznik，但仍未解决的多项式环的整数。交换代数是一个领域密切相关的代数几何：而代数几何侧重于解决方案的几何集合的多项式方程，在交换代数的主要对象的研究是功能对这些解决方案集。大多数的问题，这将是在拟议的研究中调查的问题，家庭的方程的解决方案的存在性，以及解决方案集的性质。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，密码学和其他具有战略意义的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.