Free Resolutions

免费决议

• 批准号: 1406062
• 负责人: Irena Peeva
• 金额: $ 17.75万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406062&HistoricalAwards=false
• 关键词: Free; Resolutions

A core goal in the mathematical areas Algebraic Geometry and Commutative Algebra deals with understanding the solutions of a system of polynomial equations, possibly in a large number of variables and with a large number of equations. The solutions form a geometric object. The main idea is to study the rich and beautiful interplay between its geometric and algebraic properties. Closely related to this is the concept of a free resolution, which was introduced by the famous mathematician David Hilbert in two papers in 1890 and 1893. Constructing a free resolution amounts to repeatedly solving systems of polynomial equations. The study of these objects flourished in the second half of the twentieth century, and has seen spectacular progress in the last ten years. The field is very broad, with strong connections and applications to other mathematical areas and string theory. Recent computational methods have made it possible to compute free resolutions by computer. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants.Free resolutions provide a method for describing the structure of finitely generated modules over a commutative noetherian ring. In the local and the graded cases there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution of the resolved module. Hilbert proved that every finitely generated graded module over a polynomial ring has a finite minimal free resolution. There has been a lot of progress on the properties of finite free resolutions. Much less is known about the properties of infinite free resolutions, which occur abundantly over graded non-linear quotient rings of a polynomial ring. This project deals with infinite minimal free resolutions. It focusses on complete intersections and edge rings. The objectives are: (1) to build Boij-Soderberg Theory over graded complete intersections; (2) to explore applications of matrix factorizations; (3) to construct explicit minimal free resolutions over Clements-Lindstrom rings; (4) to provide formulas for the high differentials of a minimal free resolution over a complete intersection; (5) to study infinite minimal free resolutions over edge rings. The methods to be employed in (1), (2), and (4) are those recently introduced by Eisenbud and Peeva in their work on matrix factorizations for complete intersections. (3) is a continuation of Peeva's work on Clements-Lindstrom rings. The broader impacts of the proposed activities include advising students, organizing conferences, and the PI will continue to serve as an editor of the journal Proceedings of the American Mathematical Society.

数学领域的一个核心目标是代数几何和交换代数，涉及理解多项式方程组的解，可能有大量的变量和大量的方程。解决方案形成一个几何对象。其主要思想是研究其几何和代数性质之间丰富而美丽的相互作用。与此密切相关的是自由分解的概念，这是由著名数学家大卫希尔伯特在1890年和1893年的两篇论文中介绍的。构造一个自由解就相当于反复求解多项式方程组。对这些天体的研究在20世纪后半叶蓬勃发展，在过去的十年里取得了惊人的进展。该领域非常广泛，与其他数学领域和弦理论有着密切的联系和应用。最近的计算方法使计算机计算自由分辨率成为可能。本项目的主要研究目标是在理解自由归结及其数值不变量的结构方面取得重大进展，自由归结提供了一种描述交换Noether环上的非线性生成模结构的方法。在局部和分次的情况下，存在一个最小自由分解;它在同构下是唯一的，并且包含在分解模的任何自由分解中。Hilbert证明了多项式环上的每一个n-生成分次模都有有限的极小自由分解。关于有限自由归结的性质已经有了很大的进展。关于无限自由归结的性质知之甚少，而无限自由归结在多项式环的分次非线性商环上大量出现。这个项目涉及无限的最小自由决议。它专注于完整的交叉点和边缘环。目标是：（1）建立分次完全交上的Boij-Soderberg理论;（2）探讨矩阵分解的应用;（3）构造Clements-Lindstrom环上的显式极小自由归结;（4）给出完全交上极小自由归结的高次微分公式;（5）研究边环上的无穷极小自由归结.在（1）、（2）和（4）中采用的方法是最近由Eisenbud和Peeva在他们关于完全相交的矩阵分解的工作中介绍的那些方法。(3)是Peeva关于Clements-Lindstrom环的工作的延续。拟议活动的更广泛影响包括为学生提供建议，组织会议，PI将继续担任美国数学学会学报的编辑。