A1-Homotopy Theory and Motives

Voevodsky 9901219提出者将继续构造代数簇的同伦理论。到今天为止，我们已经知道如何定义代数簇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-理论和上边。普通同伦理论的其他基本成分的类似物，如普通上同调中的Steenrod运算，也得到了部分理解。在这个阶段的项目的主要攻击将作出其他两个成分是迄今为止失踪;这些是对偶理论和识别原则的T-loop spaces.Homotopy理论的代数簇是一种方法来研究代数簇（=系统的代数方程）的基础上anananalogy。这种类比存在于仿射线的抽象范畴性质（=有一个变量和没有方程的方程组）和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簇，从而为处理代数几何和数论中的问题提供了新的工具。

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.