| On Gorenstein Homological Dimensions of Groups |
| Received:February 21, 2022 Revised:August 19, 2022 |
| Key Words:
Gorenstein homological dimension Serre's theorem group ring
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| 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). |
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| Abstract: |
| 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. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2023.02.007 |
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