整群环的Bass Nil-群

在群环的代数K-理论中,将关于任意群G的Ki(ZG)的计算归结为对关于有限群H的Ki(ZH)的研究是目前重要的问题，整群环的Bass Nil-群NKi(ZG)在这一问题上扮演着阻碍的角色，研究NKi(ZG)的结构是群环的K-理论中的核心问题之一。本项目拟研究当G是有限群时NKi(ZG)的结构问题。我们拟利用关于NK-群的M-V序列以及有限群环上的截断多项式环的K2群去证明对任意非平凡有限群G，NK2(ZG)都是非平凡的；深入研究NKi(ZG)作为Witt向量环W(Z)上的模的结构，利用Verschiebung映射、Frobenius映射等建立NKi(ZG)与NKi(ZH)之间的联系，其中H是G的p-子群或p-基本子群等，得到对扭群NKi(ZH)中元的阶数或指数的一个较为精确的估计。

In the K-theory of group rings, the Bass Nil-group NKi(ZG) are the obstruction to reduce calculations of K-groups of group rings to finite groups. In this project, we intend to study the structure of NKi(ZG) for G a finite group. Using the results on the structure of K2 groups of truncated polynomial rings over finite group rings and the Mayer-Vietoris sequence for NK-groups, we try to prove that NK2(ZG) is a non-trivial group for any finite nontrivial group G. To get a precise estimate on the exponent of the elements in the torision group NKi(ZG), we shall explore the W(Z)-module structure of NKi(ZG); using Verschiebung maps and Frobenius maps, we also intend to calculate NKi(ZG) from NKi(ZH), where H are p-subgroups or p-elementary subgroups of G.