论文标题
Gorenstein的共同体学维度和稳定类别的群体
Gorenstein cohomological dimension and stable categories for groups
论文作者
论文摘要
首先，我们研究Gorenstein的共同学维度$ {\ rm gcd} _rg $ agert $ g $ tocefliced lings $ r $，在组和戒指的变化下；给出了$ {\ rm gcd} _rg $的有限性的特征。在系数环$ \ mathbb {z} $上获得的文献或有限全球维度的一些结果被推广到更一般的情况。此外，我们在弱势的完整确切类别$ \ Mathcal {f} IB $上建立了一个模型结构，该类别由纤毛$ rg $ -modules组成，并证明同型类别$ \ mathrm {ho mathrm {ho} $ \ usepline {\ Mathcal {c} of Benson的同伴模块和稳定的模块类别$ {\ rm stmod}（rg）$。讨论了连接模块与戈伦斯坦射影模块之间的关系，我们表明，在某些条件下，$ {\ rm gcd} _rg <\ fty $，$ {\ rm ho}（\ rm ho}（\ nathcal {f}以及投影$ rg $模型的完全无环络合物的同型类别。
First we study the Gorenstein cohomological dimension ${\rm Gcd}_RG$ of groups $G$ over coefficient rings $R$, under changes of groups and rings; a characterization for finiteness of ${\rm Gcd}_RG$ is given. Some results in literature obtained over the coefficient ring $\mathbb{Z}$ or rings of finite global dimension are generalized to more general cases. Moreover, we establish a model structure on the weakly idempotent complete exact category $\mathcal{F}ib$ consisting of fibrant $RG$-modules, and show that the homotopy category $\mathrm{Ho}(\mathcal{F}ib)$ is triangle equivalent to both the stable category $\underline{\mathcal{C}of}(RG)$ of Benson's cofibrant modules, and the stable module category ${\rm StMod}(RG)$. The relation between cofibrant modules and Gorenstein projective modules is discussed, and we show that under some conditions such that ${\rm Gcd}_RG<\infty$, ${\rm Ho}(\mathcal{F}ib)$ is equivalent to the stable category of Gorenstein projective $RG$-modules, the singularity category, and the homotopy category of totally acyclic complexes of projective $RG$-modules.
