论文标题
在gorenstein群体的同源维度上
On Gorenstein homological dimension of groups
论文作者
论文摘要
让$ g $为一个组，$ r $为戒指。我们将$ g $的Gorenstein同源尺寸定义为$ r $，由$ {\ rm ghd} _ {r} g $表示为琐碎$ rg $ -module $ r $的Gorenstein Flat尺寸。证明$ {\ rm ghd} _sg \ leq {\ rm ghd} _rg $用于通勤环$ r \ rightarrow s $的任何平面扩展；特别是，$ {\ rm ghd} _ {r} g $是$ {\ rm ghd} _ {\ mathbb {z}} g $的细化，如果$ r $是$ \ mathbb {z} $ - torsion-free。我们显示了Serre定理的Gorenstein同源版本，即$ {\ rm ghd} _ {r} g = {\ rm ghd} _ {rm ghd} _ {r} h $，用于任何子组的$ h $ of $ g $ g $ a $ g $ a $ g $。作为一个应用程序，$ g $是一个有限的组，并且仅当$ {\ rm ghd} _ {r} g = 0 $;这与任何非平凡有限群体的同源维度都是无穷大的事实不同。
Let $G$ be a group and $R$ be a ring. We define the Gorenstein homological dimension of $G$ over $R$, denoted by ${\rm Ghd}_{R}G$, as the Gorenstein flat dimension of trivial $RG$-module $R$. It is proved that ${\rm Ghd}_SG \leq {\rm Ghd}_RG$ for any flat extension of commutative rings $R\rightarrow S$; in particular, ${\rm Ghd}_{R}G$ is a refinement of ${\rm Ghd}_{\mathbb{Z}}G$ if $R$ is $\mathbb{Z}$-torsion-free. We show a Gorenstein homological version of Serre's theorem, i.e. ${\rm Ghd}_{R}G = {\rm Ghd}_{R}H$ for any subgroup $H$ of $G$ with finite index. As an application, $G$ is a finite group if and only if ${\rm Ghd}_{R}G = 0$; this is different from the fact that the homological dimension of any non-trivial finite group is infinity.
