Mathematical Sciences: Multiplicative Structures on Free Resolutions & Local Algebra

数学科学：自由分辨率的乘法结构

• 批准号: 9302735
• 负责人: Hema Srinivasan
• 金额: $ 5.7万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9302735&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Multiplicative; Structures; Free

The principal investigator will study three areas of commutative algebra. The first area is concerned with multiplicative structures on free resolutions. She will find a criterion for the existence of an associative commutative differential graded algebra structure on the minimal free resolution on cyclic modules. She will also determine if such a structure exists for the minimal resolutions of the cyclic modules defined by the ideals in the linkage class of a complete intersection. The second area is concerned with fundamental groups and equivalences of singularities. In particular, she will determine a number k such that for a given determinantal ideal I in a power series ring R and for any other ideal J of the same height, J equivalent to I modulo the kth power of the Jacobean ideal of I guarantees the existence of an isomorphism of R which maps I onto J so that J is also a determinantal ideal. The third area is to determine, in positive characteristics, if the symbolic algebra of a monomial prime is finitely generated. This is reasearch in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.

首席研究员将研究三个领域， 交换代数 第一个领域涉及 自由分辨率上的乘法结构。 她会找到一个 结合可换性存在的判据 极小自由上的微分分次代数结构 循环模的分解 她还将确定， 结构存在的最小分辨率的循环 模定义的理想在一个完整的链接类 路口 第二个领域是关于基本的 群和奇点的等价。 她特别 将确定一个数k，使得对于给定的行列式， 幂级数环R中的理想I和幂级数环R中的任何其他理想J， 相同的高度，J等于I模的k次幂 I的一个理想保证了I的一个同构的存在， R将I映射到J上，使得J也是一个行列式理想。 第三个领域是确定，在积极的特点，如果 单项式素数的符号代数是可生成的。 这在代数几何领域是合理的， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。 在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。 如今，该领域 不仅使用代数方法，而且使用分析方法， 拓扑学，反过来说， 以及理论计算机科学和机器人技术。