论文标题
Gorenstein同源维度和一些群体
Gorenstein homological dimension and some invariants of groups
论文作者
论文摘要
对于任何组$ g $，Gorenstein同源尺寸$ {\ rm ghd} _rg $被定义为系数环$ r $的Gorenstein Flat dimension，这被认为是$ rg $ module，trivial Group Action。 We prove that ${\rm Ghd}_RG < \infty$ if and only if the Gorenstein flat dimension of any $RG$-module is finite, if and only if there exists an $R$-pure $RG$-monic $R\rightarrow A$ with $A$ being $R$-flat and ${\rm Ghd}_RG = {\rm fd} _ {rg} a $，其中$ r $是有限的戈伦斯坦弱全球维度的交换戒指。作为应用程序，研究了子组的$ {\ rm GHD} $的属性，商组，组的扩展以及Weyl组的扩展。此外，我们比较一些不变的人之间的关系，例如$ {\ rm sfli} rg $，$ {\ rm silf} rg $，$ {\ rm spli} rg $，$ {\ rm silp} rg $给出了与组环相比的戈伦斯坦射弹性问题的足够条件。
For any group $G$, the Gorenstein homological dimension ${\rm Ghd}_RG$ is defined to be the Gorenstein flat dimension of the coefficient ring $R$, which is considered as an $RG$-module with trivial group action. We prove that ${\rm Ghd}_RG < \infty$ if and only if the Gorenstein flat dimension of any $RG$-module is finite, if and only if there exists an $R$-pure $RG$-monic $R\rightarrow A$ with $A$ being $R$-flat and ${\rm Ghd}_RG = {\rm fd}_{RG}A$, where $R$ is a commutative ring with finite Gorenstein weak global dimension. As applications, properties of ${\rm Ghd}$ on subgroup, quotient group, extension of groups as well as Weyl group are investigated. Moreover, we compare the relations between some invariants such as ${\rm sfli}RG$, ${\rm silf}RG$, ${\rm spli}RG$, ${\rm silp}RG$, and Gorenstein projective, Gorenstein flat and PGF dimensions of $RG$-modules; a sufficient condition for Gorenstein projective-flat problem over group rings is given.
