Free Resolutions

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• 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.

数学领域代数几何和交换代数的核心目标涉及理解多项式方程组的解，可能包含大量变量和大量方程。这些解形成了一个几何对象。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辨率的概念，它是由著名数学家大卫希尔伯特在1890年和1893年的两篇论文中提出的。构造自由分辨率相当于重复求解多项式方程组。对这些物体的研究在二十世纪下半叶蓬勃发展，并在过去十年中取得了惊人的进展。该领域非常广泛，与其他数学领域和弦理论有很强的联系和应用。最近的计算方法使得通过计算机计算自由分辨率成为可能。该项目的主要研究目标是在理解自由解析结构及其数值不变量方面取得重大进展。自由解析提供了一种描述交换诺特环上有限生成模结构的方法。在本地和分级情况下，存在最小自由分辨率；它在同构上是唯一的，并且包含在已解析模块的任何自由解析中。希尔伯特证明了多项式环上的每个有限生成的分级模块都具有有限的最小自由分辨率。有限自由分辨率的性质已经取得了很多进展。人们对无限自由分辨率的性质知之甚少，无限自由分辨率在多项式环的分级非线性商环上大量出现。该项目处理无限最小自由分辨率。它专注于完整的交叉点和边缘环。目标是： (1) 在分级完全交叉点上建立 Boij-Soderberg 理论； (2)探索矩阵分解的应用； (3) 在 Clements-Lindstrom 环上构造显式最小自由分辨率； (4) 提供完整交叉点上最小自由分辨率的高微分公式； (5) 研究边缘环上的无限最小自由分辨率。 (1)、(2) 和 (4) 中使用的方法是 Eisenbud 和 Peeva 最近在完全交集的矩阵分解工作中引入的方法。 (3) 是 Peeva 对 Clements-Lindstrom 环工作的延续。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《美国数学会论文集》杂志的编辑。