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.

阿夫拉莫夫将研究交换代数、同调代数和表示论交叉点出现的问题。 将通过同调构造生成的数值、代数和几何不变量来研究交换诺特环的性质。 我们将开发新的方法来计算此类不变量，并针对传统方法无法解决的问题进行测试。 同调代数将用于在交换环理论和有限维代数理论之间建立进一步的联系，其目标是转移观点并采用在一个领域中开发的技术在另一个领域中使用。交换代数和代数几何可以被认为是研究许多未知数中的多个方程的解，通常，解不是唯一的。然后可以从几何角度查看这组解，但通常最好将其编码为在此组上定义的一系列函数。 此类函数族的抽象版本称为交换环。 同调代数带来了代数拓扑环方法的研究，该方法是为研究几何结构而开发的。 将复杂对象分解为原始构建块的方法是作为代数不同分支（称为表示论）的一部分而开发的。