Algebraic K-Theory and Motivic Cohomology

Suslin建议研究动机上同理论的主题。这些主题对于方案的上同调理论的发展至关重要，并且在Suslin和他的合作者的工作中近年来都取得了进展。首先，Suslin打算消除动机上同论基本要求中的完备性假设，并在任意域上定义相应的动机配合物的张量三角化范畴。第二个主题是试图得到一个新的、更清晰的主要对偶定理的证明，这个证明将适用于任意域上的方案（而不仅仅是像目前的证明那样适用于特征为零的域）。下一步，Suslin计划与A. Merkurjev合作，将先前关于Severi-Brauer变的动机上同的计算推广到高等符号的属分裂域的情况。也许拨款提案中最有趣和最重要的部分是试图将Morel和Levin发展的代数共调理论与Voevodsky构建的上同调理论的代数部分进行比较。这里的计划是使用Voevodsky开发的框架轮轴机械。数学的主要目标是为物理世界提供一幅准确的图画，或者至少是一幅适当的近似值。从这个角度来看，代数变量是最重要的，首先，它们相对容易理解，因为它们只是由多项式方程定义的，其次，它们通常给出了对其他形状的相当精确的近似，最重要的是，它们确实自然地出现在从理论物理到编码理论的许多学科中。这就是为什么代数几何——代数变分理论对数学的发展和应用如此重要。本项目致力于研究动机上同论的某些基本问题-一个相对较新的和非常迅速发展的代数几何分支。在这部分数学中，几何与代数和拓扑学相结合，所使用的思想和方法同样来自这三个方向。作为这项拨款提案更广泛影响的一部分，让我指出，我打算让研究生参与到这项拨款提案的某些部分的工作中，从而使他们能够进入一个快速发展且相当重要的数学领域。

Suslin proposes to investigate topics in motivic cohomology theory. These topics are of principal importance for the development of the cohomology theory of schemes and each has seen progress during the recent years in the works of Suslin and his collaborators. Firstly Suslin plans to eliminate the perfectness assumption in the basic requirements of motivic cohomology theory and to define the corresponding tensor triangulated category of motivic complexes over arbitrary fields. The second topic is an attempt to get a new and clearer proof of the main duality theorem which would work for schemes over arbitrary fields (and not only over fields of characteristic zero as the current proof does). Next Suslin plans jointly with A. Merkurjev to generalize the previous computation of the motivic cohomology of Severi-Brauer varieties to the case of generic splitting fields of higher symbols. Probably the most interesting and important part of the grant proposal is an attempt to compare the algebraic cobordism theory developed by Morel and Levin with the algebraic part of the cohomology theory constructed by Voevodsky. Here the plan is to use the machinery of framed sheaves developed by Voevodsky.The main objective of mathematics is to provide an accurate picture to the physical world or at least an appropriate approximation of that picture.From this point of view algebraic varieties are of principal importance,first they are relatively easy to understand since they are just defined by polynomial equations, next they usually give a rather accurate approximation to other shapes, most importantly they do appear naturally in quite a lot of subjects from theoretical physics to coding theory. That's why algebraic geometry - the theory of algebraic varieties is so important for the development and applications of mathematics. This project is devoted to the study of certain fundamental problems of motivic cohomology theory - a relatively new and very quickly developing branch of algebraic geometry. Geometry is blended with algebra and topology in this part of mathematics, ideas and methods to be used come equally from all these directions. As part of the broader impact of this grant proposal let me point out that I intend to involve graduate students into the work over some parts of this grant proposal thus allowing them to get into a fast developing and quite important field of mathematics.