Free Resolutions

在数学领域，代数几何和交换代数的一个核心目标是了解多项式方程组的解，可能是在大量变量和大量方程中。这些解形成了一个几何对象。其主要思想是研究它的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解的概念，这一概念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。构造自由解相当于重复求解多项式方程组。对这些物体的研究在20世纪下半叶蓬勃发展，并在最近十年中取得了令人瞩目的进展。这个领域非常广泛，与其他数学领域和弦理论有着很强的联系和应用。最近的计算方法使用计算机计算自由分辨率成为可能。这个项目的主要研究目标是在理解自由分辨率及其数值不变量方面取得重大进展。自由分辨率提供了一种描述交换Notherian环上有限生成模的结构的方法。在局部和分次情形下，存在一个最小自由分解；它在同构之前是唯一的，并且包含在分解的模的任何自由分解中。Hilbert证明了多项式环上的每个有限生成分次模都有一个有限极小自由分解。关于有限自由分解的性质已经有了很大的进展。关于无限自由分解的性质，我们所知的要少得多，这种性质大量地出现在多项式环的分次非线性商环上。这个项目处理无限最小自由分辨率。它专注于完全交集和边环。其目的是：(1)建立分次完全交上的BOIJ-Soderberg理论；(2)探索矩阵分解的应用；(3)构造Clements-Lindstrom环上的显式极小自由分解；(4)给出完全交上极小自由分解的高微分公式；(5)研究边环上的无限极小自由分解.(1)、(2)和(4)中使用的方法是Eisenbud和Peeva最近在他们关于完全交的矩阵因式分解的工作中介绍的方法。(3)是Peeva关于Clements-Lindstrom环的工作的继续。拟议活动的更广泛影响包括为学生提供建议、组织会议，国际数学联合会将继续担任《美国数学学会学报》杂志的编辑。

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.