K-Theory and Motivic Cohomology

9876729本建议涉及动机上同的各个方面及其与k理论和Hodge理论的关系的研究。一个项目是将Bloch的高Chow群理论推广到混合特征的格式，并构造一个从高Chow群到该格式上相干束范畴的k理论的谱序列。本项目的进一步目标是建立高周群的预期性质，为正则格式提供一个良好的动机上同调理论，以及上述谱序列的预期性质。第二个项目是用动机上同调的全纯版本作为模理论的替代，给出Beilinson-Lichtenbaum猜想的积分版本。第三个课题是研究具有对数极点和积分周期的微分形式的一些模性质，第四个课题是给出经典群的积分动机的计算。这项研究属于数论领域。数论的历史根源在于对自然数的研究。它是数学最古老的分支之一。在过去的半个世纪里，它已经成为数据传输和处理以及通信系统等各种应用领域不可或缺的工具。

9876729The proposal involves a study of various aspects of motivic cohomology and its relationship with K-theory and Hodge theory. One project is to extend the theory of Bloch's higher Chow groups to schemes in mixed characteristic and to construct a spectral sequence from the higher Chow groups to the K-theory of the category of coherent sheaves on such a scheme. A further goal of this project is to establish the expected properties of the higher Chow groups to furnish a good theory of motivic cohomology for regular schemes, as well as the expected properties of the above-mentioned spectral sequence. A second project is to give an integral version of the Beilinson-Lichtenbaum conjectures using a holomorphic version of motivic cohomology as a replacement for the modular theory. A third project is to investigate some modular properties of differential forms with log poles and integral periods, and the fourth project is to give a computation of the integral motive for classical groups.This research is in the area of number theory. Number theory has its historical roots in the study of natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensible tool in diverse applications in areas such as data transmission and processing, and communication systems.