On Gorenstein Homological Dimensions of Groups
Received:February 21, 2022  Revised:August 19, 2022
Key Words: Gorenstein homological dimension   Serre's theorem   group ring  
Fund Project:Supported by the Natural Science Foundation of Hunan Province (Grant No.2021JJ30536) and the Scientific Research Foundation of Hunan Provincial Education Department (Grant No.21A0493).
Author NameAffiliation
Yueming XIANG School of Mathematics and Computational Science, Huaihua University, Hunan 418000, P. R. China 
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      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.
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