Syzygies

赛兹吉斯

• 批准号: 1100046
• 负责人: Irena Peeva
• 金额: $ 21.85万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-01 至 2014-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1100046&HistoricalAwards=false
• 关键词: Syzygies

This project is in the field of commutative algebra. The main research goal is to understand the structure of minimal free resolutions over complete intersection ring. The idea is to describe the asymptotic structure of the generic Eisenbud operators, and then use it in order to obtain the differentials. The proposed research also includes a topic involving interdisciplinary approaches connecting commutative algebra with the fields of combinatorics and topology. The PI will co-organize two conferences in commutative algebra, which will provide a forum for discussion of recent developments and foster research in new directions.Research on free resolutions is a core and beautiful area in commutative algebra. It contains a number of challenging and important conjectures and open problems. The idea to associate a free resolution to a finitely generated module was introduced by Hilbert in two famous papers in 1890 and 1893. He proved that over a polynomial ring (over a field) every finitely generated module has a finite resolution. 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. Its properties are closely related to the invariants of the resolved module. For many years, minimal free resolutions have been both central objects and fruitful tools in Commutative Algebra; they also have applications in other mathematical fields.

该项目属于交换代数领域。主要研究目标是了解完整相交环上最小自由分辨率的结构。这个想法是描述通用 Eisenbud 算子的渐近结构，然后使用它来获得微分。拟议的研究还包括涉及将交换代数与组合学和拓扑学领域连接起来的跨学科方法的主题。 PI将共同组织两场交换代数会议，这将为讨论最近的发展提供一个论坛，并促进新方向的研究。自由解析研究是交换代数的核心和美丽的领域。它包含许多具有挑战性的重要猜想和未解决的问题。将自由分辨率与有限生成模块相关联的想法是由希尔伯特在 1890 年和 1893 年的两篇著名论文中提出的。他证明了在多项式环上（在一个域上），每个有限生成模块都具有有限分辨率。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。它的属性与解析模块的不变量密切相关。多年来，最小自由分辨率一直是交换代数的中心对象和富有成效的工具。它们在其他数学领域也有应用。