Algebraic K-Theory and Motivic Cohomology

Suslin提出了动机上同调理论的研究课题。这些议题是最重要的发展上同调理论的计划和每一个已经看到了进展，在最近几年的作品苏斯林和他的合作者。首先，Suslin计划消除motivic上同调理论基本要求中的完全性假设，并定义任意域上motivic复形的相应张量三角范畴。第二个主题是试图得到一个新的和更清楚的证明的主要对偶定理，将工作计划在任意领域（而不仅仅是在外地的特征零作为目前的证明）。接下来，Suslin与A. Merkurjev推广以前的计算动机上同调的Severi-Brauer品种的情况下，一般分裂领域的更高的符号。可能是最有趣和重要的部分拨款建议是试图比较代数cobordism理论发展的莫雷尔和莱文与代数的一部分，上同调理论建设Voevodsky。这里的计划是使用机械的框架层开发的Voevodsky。数学的主要目标是提供一个准确的图片，以物理世界或至少是一个适当的近似，从这个角度来看，代数簇是主要的重要性，首先，他们是相对容易理解，因为他们只是定义的多项式方程，其次，它们通常给出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相当多的学科中。这就是为什么代数几何-代数簇的理论对数学的发展和应用如此重要。这个项目致力于研究动机上同调理论的某些基本问题--这是代数几何学中一个相对较新且发展很快的分支。在数学的这一部分中，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所有这些方向。作为更广泛的影响，这项拨款建议的一部分，让我指出，我打算让研究生参与工作的某些部分，这项拨款建议，从而使他们能够进入一个快速发展和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.