Studies in Commutative Algebra

PI将研究交换代数中的几个问题，这些问题围绕着同调代数，紧闭包和局部上同调。其中一些，如同调代数，是老问题，最近的进展提供了解决的希望;紧闭和有限性的问题局部上同调模是PI的长期项目的延续。霍希斯特的单项猜想是解决环不包含一个领域，如那些出现在数论。两种方法，这将是追求：第一个是一个自然的延伸海特曼的工作，另一个是通过局部上同调理论。Brenner和Monsky最近证明了紧密封闭不需要与本地化互换。然而，似乎弱F正则性--环的所有理想都是紧闭的性质--确实局部化了。这将通过分裂环的概念来处理。本文还对Lyubeznik关于正则环的局部上同调模有1000个相伴素理想的猜想提出了攻击。这在各种情况下都是已知的，但是对于整数上的多项式环仍然没有解决。这个项目关注的是交换代数中的问题。这是一个与代数几何密切相关的领域：代数几何专注于多项式方程的解集的几何，而交换代数的观点是研究解集上多项式函数的环。大多数的问题，将被调查可能被视为问题的存在性的解决方案的家庭方程，以及有关的性质的解决方案集。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 PI will work on several problems in commutative algebra that are centered around the homological conjectures, tight closure, and local cohomology. Some of these, such as the homological conjectures, are old problems for which recent advances have provided the hope of a solution; the problems on tight closure and finiteness properties of local cohomology modules are a continuation of the PI's long-term projects. Hochster's monomial conjecture is unresolved for rings that do not contain a field, such as those arising in number theory. Two approaches to this will be pursued: the first is a natural extension of Heitmann's work; another is via local cohomology theory. Brenner and Monsky recently proved that tight closure need not commute with localization. However, it appears likely that weak F-regularity---the property that all ideals of a ring are tightly closed---does localize. This will be approached via the notion of splinter rings. It is also proposed to attack Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases, but remains unresolved for polynomial rings over the integers.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 may be viewed as 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.