Algebraic K-theory and Equivariant Homotopy Theory

代数K理论和等变同伦理论

• 批准号: 1007083
• 负责人: Teena Gerhardt
• 金额: $ 10.39万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007083&HistoricalAwards=false
• 关键词: Algebraic; theory; Equivariant; Homotopy; Theory

The theme of this project is to use the tools of equivariant stable homotopy theory to study algebraic K-theory, particularly the K-theory of singular and filtered rings. Although the definition of algebraic K-theory is not inherently equivariant, the tools of equivariant stable homotopy theory have proven useful for K-theory computations. In particular, one fruitful approach exploits the equivariant structure of topological Hochschild homology (THH) to compute algebraic K-theory. In the case of certain singular rings, this approach reduces the computation of K-theory to the computation of equivariant stable homotopy groups of THH, graded by the real representation ring of the circle. To compute K-theory one needs to determine which equivariant homotopy groups arise, compute these groups, and then assemble them to recover K-theory. Each of these steps is difficult and understood only in a small number of cases. This project seeks to address these issues for various specific K-theory computations, as well as defining an abstract algebraic object embodying structures that arise in these computations. Other specific goals of the project include developing an approach for the K-theory of filtered rings, and answering several questions about the structure of THH. Algebraic K-theory is an invariant which can be applied to study basic objects from several fields of mathematics. In particular, algebraic K-theory can be used to study properties of fundamental objects in algebra, called rings. Although higher algebraic K-theory was defined more than 30 years ago, computational progress has been slow. Indeed, even for some very basic rings, the algebraic K-theory is still not known. K-theory computations, however, have important applications to many areas of mathematics: algebraic number theory, classification of manifolds, motivic homotopy theory, special values of L-functions, etc. An approach to these important computations lies in the field of algebraic topology, and more specifically, in the study of equivariant homotopy theory. The goal of this project is to use these tools to not only produce new algebraic K-theory computations, but also to develop the framework and theory to facilitate future computations.

本项目的主题是利用等变稳定同伦理论的工具来研究代数K理论，特别是奇异环和滤波环的K理论。尽管代数 K 理论的定义本质上不是等变的，但等变稳定同伦理论的工具已被证明对于 K 理论计算很有用。特别是，一种卓有成效的方法利用拓扑 Hochschild 同调 (THH) 的等变结构来计算代数 K 理论。在某些奇异环的情况下，该方法将 K 理论的计算简化为 THH 的等变稳定同伦群的计算，由圆的实表示环分级。为了计算 K 理论，我们需要确定出现哪些等变同伦群，计算这些群，然后将它们组合起来以恢复 K 理论。这些步骤中的每一步都很困难，并且仅在少数情况下才能理解。该项目旨在解决各种特定 K 理论计算的这些问题，并定义一个体现这些计算中出现的结构的抽象代数对象。该项目的其他具体目标包括开发过滤环 K 理论的方法，并回答有关 THH 结构的几个问题。代数 K 理论是一个不变量，可应用于研究多个数学领域的基本对象。特别是，代数 K 理论可用于研究代数中称为环的基本对象的属性。尽管高等代数 K 理论在 30 多年前就已定义，但计算进展缓慢。事实上，即使对于一些非常基本的环，代数 K 理论仍然未知。然而，K 理论计算在许多数学领域都有重要的应用：代数数论、流形分类、动机同伦理论、L 函数的特殊值等。这些重要计算的方法在于代数拓扑领域，更具体地说，是在等变同伦理论的研究中。该项目的目标是使用这些工具不仅产生新的代数 K 理论计算，而且开发框架和理论以促进未来的计算。