Free Resolutions in Commutative Algebra

交换代数中的自由解析

• 批准号: 1702125
• 负责人: Irena Peeva
• 金额: $ 19.2万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-01 至 2021-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1702125&HistoricalAwards=false
• 关键词: Free; Resolutions; Commutative; Algebra

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 study is the concept of a free resolution, which was first introduced by 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 some free resolutions by computers. The main research goal in this project is to make significant progress in understanding the structure of free resolutions and their numerical invariants. The main research topics are:(1) resolutions over complete intersections, which will be studied using the methods recently introduced by Eisenbud and Peeva in their research monograph "Minimal Free Resolutions over Complete Intersections";(2) Betti numbers of periodic infinite minimal free resolutions, for which computational algebra methods will be combined with insights from the examples of modules with periodic resolutions with constant Betti numbers;(3) applications of the new approach introduced recently by McCullough and Peeva which produces resolutions of prime ideals.

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。与这项研究密切相关的是自由分辨率的概念，它由 David Hilbert 在 1890 年和 1893 年的两篇论文中首次提出。构建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计算机计算一些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分辨率的结构及其数值不变量方面取得重大进展。主要研究课题为：（1）完全交集上的解析，采用Eisenbud和Peeva最近在其研究专着《完全交集上的最小自由解析》中介绍的方法进行研究；（2）周期无限最小自由的Betti数分辨率，计算代数方法将与来自具有恒定贝蒂数的周期性分辨率的模块示例的见解相结合；(3) McCullough 和 Peeva 最近引入的新方法的应用，该方法产生素数的分辨率理想。

项目成果

Regularity of prime ideals

素理想的正则性

• DOI: 10.1007/s00209-018-2089-y
• 发表时间: 2019
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Caviglia, Giulio;Chardin, Marc;McCullough, Jason;Peeva, Irena;Varbaro, Matteo
• 通讯作者: Varbaro, Matteo

Non-commutative CI operators

非交换 CI 运算符

• DOI: 10.1090/proc/14480
• 发表时间: 2019
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Eisenbud, David;Peeva, Irena;Schreyer, Frank-Olaf
• 通讯作者: Schreyer, Frank-Olaf

Minimal Resolutions Over Codimension 2 Complete Intersections

余维 2 完整交集的最小分辨率

• DOI: 10.1007/s40306-018-0293-9
• 发表时间: 2019
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: Eisenbud, David;Peeva, Irena
• 通讯作者: Peeva, Irena