有限交换群代数的K2群

The calculation of the K-group of group ring is one of the most important subjects in algebraic K-theory, and in this project we will study the structure of K2(FG) and K2(ZG/(p^r)). We show the concrete structure of these K2-groups through delicately analysing the generators that are represented by Dennis-Stein symbols or Steinberg symbols of these finite abelian groups, and then use Mayer-Vietoris sequence and the relative long exact sequence in K-theory. The main contents of our research are as follows: totally settle the problem on the structure of the K2-group of finite abelian group algebra FG with F a finite field; show the concrete structure of K2(ZG/(p^r)) for G an elementary abelian p-group, a cyclic homogeneous p-group,etc. All these research works will make great effect on developing and enriching the theory on K2-group of finite abelian group ring, and will also make great sense on the research of K2(ZG).

计算群环的K-群是代数K-理论的重要研究内容，本项目主要研究K2（FG）及K2（ZG/(p^r)）的结构问题。我们通过细致分析生成这些有限交换群的Dennis-Stein符号或Steinberg符号，利用相关的Mayer-Vietoris序列及相对正合列等，揭示这些K2群的具体结构。主要研究内容为：完全解决任意有限域F上有限交换群代数FG的K2群的具体结构问题；对基本交换p-群、循环齐次p-群等，揭示K2（ZG/(p^r）)的具体结构。这些工作将大大发展和充实有限交换群环的K2群的理论，对K2（ZG）的研究具有重要的意义。

本项目主要研究有限交换群代数的K2-群的结构问题，基本完成了申请书中的拟解决的问题。对于特征为p的有限域F和任意有限交换p-群G，首先我们通过细致分析和研究K2(FG[t]/(t^{p^m}),(t))的生成元之间的关系确定了FG上截断多项式环的相对K2群的具体结构，并最终得到了当G为任意有限交换群时有限交换群代数FG的K2群的具体结构，彻底解决了K2(FG)的结构问题。