Studies in Commutative Algebra and Algebraic Geometry

交换代数和代数几何研究

• 批准号: 1401384
• 负责人: Melvin Hochster
• 金额: $ 45.38万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401384&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra; Algebraic; Geometry

The PI plans to work on several questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of many polynomial equations in many variables. Understanding this problem is of fundamental importance in many sciences, in engineering, and in other disciplines as well. It is often difficult to determine whether to expect any solution, finitely many solutions, or infinitely many: in the last case one wants to know how many degrees of freedom one has in describing the solutions. One can study the solutions geometrically, or algebraically, by investigating certain functions on the solution space that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry that is very valuable. The problems proposed for study are long standing and of fundamental, central importance. The results obtained will be disseminated by journal and book publication, lectures at conferences and workshops, and via the internet. There is a strong educational component. The principal investigator has had thirty-nine Ph.D. students including fourteen women (and has five other Ph.D. students currently, including four women, one of whom is African-American), and has served as mentor to fourteen junior faculty members, of whom six were women. This level of activity will continue. The PI will integrate undergraduate students into his research.The PI will investigate several long standing questions in the theory Noetherian rings. One is to prove Stillman's conjecture bounding projective dimension, which Tigran Ananyan and Hochster have done in characteristic not 2, 3 for degree at most four. Another is to continue the development of tight closure theory: Neil Epstein and Hochster have developed a new version with many of the properties of the original theory that gives a smaller closure, is defined in both characteristic p and equal characteristic 0, and commutes with localization. This theory raises new questions while offering insight into existing ones. Jointly with Bhargav Bhatt, Hochster has given a new proof of the positivity of Serre intersection multiplicities over regular rings in positive characteristic using an idea that has promise for solving the more than fifty year old problem of settling the general case in mixed characteristic. Other directions include the relatively recent theory of quasilength and content of local cohomology, finiteness of support of local cohomology, Lech's conjecture and generalizations, a new approach to the long standing direct summand conjecture, and the geometry of certain algebraic sets associated with matrices. A number of these problems are intended for collaboration with graduate students and postdoctoral faculty.

PI计划研究交换代数和代数几何中的几个问题。 代数几何学研究多变量多项式方程的解。 理解这个问题在许多科学、工程和其他学科中具有根本的重要性。通常很难确定是否期望有任何解、无穷多个解或无穷多个解：在最后一种情况下，人们想知道在描述解时有多少自由度。人们可以通过研究形成所谓交换环的解空间上的某些函数来几何地或代数地研究解。 这种双重视角在交换代数和代数几何之间建立了非常有价值的密切联系。所提出的研究问题是长期存在的，具有根本性和中心重要性。 所取得的成果将通过期刊和书籍出版物、会议和讲习班上的讲座以及通过互联网传播。有很强的教育成分。首席研究员拥有39个博士学位。学生包括14名妇女（并有五个其他博士学位。目前，该学院有10名学生，包括4名妇女，其中一名是非洲裔美国人，并担任14名初级教员的导师，其中6名是妇女。 这种活动水平将继续下去。 PI将整合本科生到他的研究。PI将调查在理论诺特环几个长期存在的问题。一是证明了Tigran Ananyan和Hochster在特征不为2，3且次数至多为4的情况下所做的Stillman猜想的投影维数界。 另一个是继续发展紧闭理论： Neil Epstein和Hochster开发了一个新的版本，它具有原始理论的许多性质，给出了一个更小的闭包，定义在特征p和相等特征0中，并与局部化互换。这一理论提出了新的问题，同时提供了对现有问题的见解。与Bhargav Bhatt一起，Hochster给出了一个新的证明，证明了正则环上正特征的Serre交叉重数的正性，使用了一个想法，这个想法有希望解决五十多年来解决混合特征的一般情况的问题。 其他方向包括相对较新的理论quasilength和内容的地方上同调，有限的支持地方上同调，莱赫的猜想和推广，一种新的方法长期存在的直接和项猜想，以及几何的某些代数集与矩阵。其中一些问题是打算与研究生和博士后教师合作。