Mathematical Sciences: Homological Questions in Commutative Algebra

数学科学：交换代数中的同调问题

• 批准号: 9502819
• 负责人: Paul Roberts
• 金额: $ 15.62万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9502819&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Homological; Questions; Commutative

This award supports the continued study of certain invariants, known as local Chern characters, defined for modules over Noetherian rings. While numerous properties of these invariants have recently been discovered, there are still many open questions. The main line of study concerns invariants defined by matrices in a free resolution of a module over a local ring, where relations between these invariants and classical results in commutative algebra have led to new interpretations of them, which, although they are still conjectural in many cases, give much easier methods for their computation. Another line of research concerns the details of the construction itself, and gives alternative definitions for intersection multiplicities, leading, among other things, to a new approach to Serre's positivity conjecture. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.

该奖项支持继续研究某些不变量，称为本地陈特征，定义为诺特环上的模。 虽然最近发现了这些不变量的许多性质，但仍然有许多悬而未决的问题。 研究的主线关注由局部环上模的自由分解中的矩阵定义的不变量，这些不变量与交换代数中的经典结果之间的关系导致了对它们的新解释，尽管它们在许多情况下仍然是抽象的，但它们的计算方法要容易得多。 另一条研究路线关注的是结构本身的细节，并给出了交叉多重性的替代定义，导致了塞尔的正性猜想的新方法。 本研究涉及交换代数和代数几何中的一些问题。 代数几何研究多项式方程族的解。 人们既可以研究解集的几何，也可以通过研究形成交换环的解集上的某些函数来代数地处理问题。 这种双重视角在交换代数和代数几何之间建立了密切的联系。