群扩张的Gorenstein同调维数
Gorenstein homological dimension of group extensions
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中文总结 AI 辅助
该研究针对交换环k上FP_∞型群，证明其三类自然维数一致，建立Fel'dman定理的Gorenstein同调类似结果等，拓展了群同调理论相关结论。
中文摘要 AI 辅助
我们研究交换环k上FP_∞型群G的Gorenstein同调维数Ghd_k G。主要技术结果表明，当环R的Gorenstein弱整体维数有限时，每个有限表现的Gorenstein平坦R-模都是投射余解Gorenstein平坦的。因此，对于k上FP_∞型且sfli k<∞的群G，其三个自然维数——Gorenstein同调维数、Gorenstein上同调维数和投射余解Gorenstein平坦维数——完全一致。基于该结论，我们建立了Fel'dman定理的Gorenstein同调类似结果、Z上FP_∞型群N的m次自扩张公式，以及域检测定理，证明主理想整环上FP_∞型群的Gorenstein同调维数可通过过渡到合适的域来实现。
英文摘要
We study the Gorenstein homological dimension $Ghd_k G$ of groups $G$ which are of type $FP_{\infty}$ over a commutative ring $k$. Our main result shows that, over an arbitrary ring $R$, every finitely presented Gorenstein flat $R$-module is projectively coresolved Gorenstein flat. Consequently, for a group $G$ of type $FP_\infty$ over $k$ with sfli$k<\infty$, the three natural dimensions for $G$, namely Gorenstein homological, Gorenstein cohomological and projectively coresolved Gorenstein flat, all coincide. Building on this collapse, we establish a Gorenstein homological analogue of Fel$'$dman's theorem, a formula for iterated $m$-fold self-extensions of a group $N$ of type $FP_\infty$ over $\mathbb Z$, and a field-detection theorem, showing that the Gorenstein homological dimension of a group of type $FP_\infty$ over a principal ideal domain is realized after passing to a suitable field.
发表机构
- University of Thessaly（色萨利大学）
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