群的Gorenstein同调维数

On Gorenstein Homological Dimensions of Groups

中文关键词:  Gorenstein同调维数  Serre定理  群环

英文关键词:Gorenstein homological dimension  Serre's theorem  group ring

中文摘要:

      设$k$是一个弱维数有限的交换环, $G$是一个群. 本文讨论了群$G$具有有限的Gorenstein同调维数的标准.证明了群$G$的Gorenstein同调维数的有限性与群环$kG$的Gorenstein弱维数的有限性是一致的.进一步,我们给出了Serre定理的一个Gorenstein类比.推广了整环上$G$的Gorenstein同调维数的一些已知结果.

英文摘要:

      Let $k$ be a commutative ring with finite weak dimension and let $G$ be a group. In this paper, we explore the criterion that a group $G$ has finite Gorenstein homological dimension. It is shown that the finiteness of the Gorenstein homological dimension of $G$ coincides with the finiteness of the Gorenstein weak dimension of the group ring $kG$. Furthermore, we give a Gorenstein analogy of the Serre's theorem. Some well-known results for the Gorenstein homological dimension of $G$ over the integer ring are also extended.