Free Resolutions

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。该项目属于交换代数领域。提出的研究涉及分级自由分辨率的结构、它们的数值不变量及其应用。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解。该项目的重点是：Clements-Lindstrom环上映射锥的Borel理想及其应用，单项边理想的分辨率，以及无限胞元分辨率。所提出的研究涉及跨学科的方法，并将交换代数与组合学、计算代数和拓扑学领域联系起来。Hilbert在1890年和1893年的两篇著名论文中提出了将自由归结与有限生成模联系起来的想法。他证明了在多项式环(域上)上，每个有限生成模都有一个有限自由分解。如果环和模是分次的，则存在一个最小自由分解；它在同构之前是唯一的，并且包含在任何自由分解中。最小自由分辨是分次的，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，构造自由分解包括重复求解线性方程组。最近的计算方法使用计算机计算分级自由分辨率成为可能。多年来，自由分解一直是交换代数的中心对象和卓有成效的工具，它们在其他数学领域有着广泛的应用。

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.