A1-Homotopy Theory and Motives

A1-同伦理论和动机

• 批准号: 9901219
• 负责人: Vladimir Voevodsky
• 金额: $ 14.15万
• 依托单位: Institute For Advanced Study
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9901219&HistoricalAwards=false
• 关键词: A1; Homotopy; Theory; Motives

Voevodsky9901219The proposer will continue to construct the homotopy theory ofalgebraic varieties. As of today we know how to define the stable andunstable homotopy categories for algebraic varieties and thefundamental cohomology theories such as the ordinary (=motivic)cohomology, K-theory and cobordisms. The analogs of other basic ingredients of theordinary homotopy theory, such as the Steenrod operations in ordinarycohomology, are also partly understood. During this stage of the project the main attack will be made on two other ingredients which are so far missing; these are the duality theory and the recognition principle for T-loop spaces.Homotopy theory of algebraic varieties is an approach to the study ofalgebraic varieties (= systems of algebraic equations) based on ananalogy. This analogy exists between the abstract categoricalproperties of the affine line (= the system of equations with onevariable and no equations) and the toplogical space [0,1] called theunit interval. Using it one can translate methods of homotopy theory from topological spaces to algebraic varieties which gives new tools to deal with problems in algebraic geometry and number theory.

作者将继续建构代数变异的同伦理论。到目前为止，我们已经知道如何定义代数变异的稳定和不稳定同伦范畴，以及基本的上同调理论，如普通上同调、k理论和协群。普通同伦理论的其他基本成分的类似物，如普通上同伦中的Steenrod运算，也被部分地理解。在项目的这个阶段，主要的攻击将集中在另外两个迄今为止缺少的要素上；这就是对偶理论和t环空间的识别原理。代数变量的同伦理论是一种基于类比的研究代数变量（=代数方程组）的方法。这种类比存在于仿射线（=一元无方程方程组）的抽象范畴性质与称为单位区间的拓扑空间[0,1]之间。利用它可以将同伦理论的方法从拓扑空间转化为代数变量，从而为处理代数几何和数论问题提供了新的工具。