Free Resolutions

免费决议

• 批准号: 0900931
• 负责人: Irena Peeva
• 金额: $ 10.75万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0900931&HistoricalAwards=false
• 关键词: Free; Resolutions

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is in the field of Commutative Algebra. The proposed research deals with the structure of graded free resolutions, their numerical invariants, and applications. The general research goal is to study graded free resolutions using a variety of methods from Commutative Algebra, Computational Algebra, and Topological Combinatorics. The project focuses on: Borel ideals and applications of mapping cones over Clements-Lindstrom rings, resolutions of monomial edge ideals, and infinite cellular resolutions. The proposed research involves interdisciplinary approaches and connects Commutative Algebra with the fields of Combinatorics, Computational Algebra, and Topology.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 free resolution. If the ring and the module are graded then there exists a minimal free resolution; it is unique up to an isomorphism and is contained in any free resolution. The minimal free resolution is graded, and its properties are closely related to the invariants of the module. From another point of view: in essence constructing a free resolution consists of repeatedly solving systems of linear equations. Recent computational methods have made it possible to compute graded free resolutions by computer. For many years, free resolutions have been both central objects and fruitful tools in Commutative Algebra; they have many applications in other mathematical fields.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。该项目是在交换代数领域。建议的研究涉及结构的分级自由决议，其数值不变量，和应用。一般的研究目标是使用交换代数，计算代数和拓扑组合学的各种方法来研究分级自由分辨率。该项目的重点是：Borel理想和Clements-Lindstrom环上映射锥的应用，单项边理想的分解和无限胞腔分解。希尔伯特在1890年和1893年的两篇著名论文中提出了将自由分解与可交换生成模联系起来的思想。他证明，在一个多项式环（在一个领域）的每一个numbergenerated模块有一个有限的自由决议。如果环和模是分次的，则存在一个极小自由分解;它在同构下是唯一的，并且包含在任何自由分解中。最小自由归结是分次的，它的性质与模的不变量密切相关。从另一个角度来看：在本质上，构建一个自由的解决方案包括反复求解线性方程组。最近的计算方法使计算机计算分级自由分辨率成为可能。多年来，自由归结一直是交换代数的中心对象和卓有成效的工具，在其他数学领域也有许多应用。