Studies in Commutative Algebra

交换代数研究

• 批准号: 0070268
• 负责人: Anurag Singh
• 金额: $ 7.41万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070268&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra

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.

调查员将工作的一些问题，在理论上的交换环所产生的紧封闭理论，理论的交叉多重性，研究当地的上同调模，并从同调代数的地方环。在紧闭包理论中，有一个实质性的重点是理解F-正则环类和发展这类环的定理。 局部上同调模的研究主要涉及混合特征正则环上局部上同调模的有限性问题，以及与实闭包理论相关的另一个问题。 关于希尔伯特-昆兹多重性和托尔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：代数几何专注于多项式方程的解集的几何，而交换代数的主要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是由于革命性的工作格罗滕迪克，塞尔和Zurkki。 一个活跃的研究领域，交换代数今天是理论的紧封闭发展的Hochster和Huneke。 这个理论为现有的定理提供了更强的表述，并将许多看似无关的问题结合在一起。它也与几何物体的奇点研究有着密切的联系。 局部上同调理论是由格罗滕迪克发展起来的，他用它获得了几个惊人的结果。 局部上同调可以应用于一些基本的，但又很深的问题，例如确定定义一个代数集所需的多项式方程的最小数目。 它继续发展一个迷人的互动与其他几个数学分支。