Cohomology over Commutative Rings: Structure and Applications

交换环上同调：结构和应用

• 批准号: 1103176
• 负责人: Luchezar Avramov
• 金额: $ 45.89万
• 依托单位: University of Nebraska-Lincoln
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2019-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103176&HistoricalAwards=false
• 关键词: Cohomology; over; Commutative; Rings; Structure

Avramov will investigate problems arising at the crossroads of commutative algebra, homological algebra, and representation theory. Properties of commutative noetherian rings will be studied through numerical, algebraic, and geometric invariants generated by homological constructions. New methods will be developed for computing such invariants, to be tested on problems that have resisted more conventional approaches. Homological algebra will be utilized to establish further connections between the theories of commutative rings and of finite-dimensional algebras, with the goal of transferring viewpoints and adapting techiques developed in one of the fields for use in the other. Commutative algebra and algebraic geometry may be thought of as studying solutions of several equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but it is often best encoded into a family of functions defined on this set. The abstract version of such families of functions are called commutative rings. Homological algebra brings to the study of rings methods of algebraic topology, developed to study geometric structures. Methods for decomposing complicated objects into primitive building blocks are developed as part of a different branch of algebra, known as representation theory.

阿夫拉莫夫将调查交换代数，同调代数和表示论的十字路口出现的问题。 交换诺特环的性质将通过同调结构产生的数值、代数和几何不变量来研究。 新的方法将被开发用于计算这样的不变量，以测试的问题，抵制更传统的方法。 同调代数将被用来建立交换环和有限维代数理论之间的进一步联系，目的是转移观点和适应在一个领域开发的技术用于其他领域。交换代数和代数几何可以被认为是研究多个方程在许多未知数下的解，通常情况下，解不是唯一的。解的集合可以被几何地观察，但它通常最好被编码到定义在这个集合上的函数族中。 此类函数族的抽象版本称为交换环。 同调代数带来了研究环的方法代数拓扑，发展研究几何结构。 将复杂对象分解为基本构建块的方法是作为代数的另一个分支（称为表示论）的一部分而发展起来的。