Studies in Commutative Algebra

本文将研究交换环理论中的一些问题，这些问题源于紧闭包理论、交重数理论、局部上同调模的研究以及局部环的同调猜想。在紧闭包理论中，对F-正则环类的理解和关于这类环的定理的发展是非常重要的。局部上同调模的研究解决了混合特征正则环上的局部上同调模的有限性问题，以及与固体闭包理论有关的另一个问题。关于希尔伯特-昆兹多重性和Tor函子刚性的研究将继续与克劳迪娅·米勒合作进行。交换代数是一个与代数几何密切相关的领域：代数几何主要研究多项式方程的解集的几何，而在交换代数中，主要的研究对象是这些解集上的某些函数。代数和几何之间的联系在很大程度上要归功于格罗森迪克、瑟尔和扎里斯基的革命性工作。当今交换代数的一个活跃研究领域是由Hochster和Huneke提出的紧闭包理论。这一理论为现有定理提供了更强有力的表述，并将许多看似无关的问题结合在一起。它还与几何对象奇点的研究有很强的联系。局部上同调理论是由Grothendieck发展起来的，他利用局部上同调理论得到了几个引人注目的结果。局部上同调在一些基本而深刻的问题上有应用，例如确定定义一个代数集合所需的最小多项式的数目。它继续与其他几个数学分支发展着令人着迷的互动。

The investigator will work on some questions in the theory of commutative rings that arise from tight closure theory, the theory of intersection multiplicities, a study of local cohomology modules, and from the homological conjectures for local rings. In tight closure theory, there is a substantial focus on understanding the class of F-regular rings and on developing theorems for this class of rings. The study of local cohomology modules addresses finiteness issues of these modules over regular rings of mixed characteristic, and another question that is related to the theory of solid closure. Research on Hilbert-Kunz multiplicities and on the rigidity of the Tor functor will be carried out in continued collaboration with Claudia Miller. 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 certain functions on these solution sets. The connections between algebra and geometry are largely due to the revolutionary work of Grothendieck, Serre and Zariski. An active area of research in commutative algebra today is the theory of tight closure developed by Hochster and Huneke. This theory provides stronger formulations of existing theorems, and brings together many seemingly unrelated problems. It also has strong connections with the study of singularities of geometric objects. The theory of local cohomology was developed by Grothendieck who used it to obtain several striking results. Local cohomology has applications to basic, and yet deep, questions such as determining the minimal number of polynomial equations needed to define an algebraic set. It continues to develop a fascinating interaction with several other branches of mathematics.