交叉积的同调维数
Homological dimensions of crossed products
摘要：
本文考虑交叉积$A _α ^σ G$的几个同调维数，其中$A$是左诺特环，$G$是有限群.我们重温了导出范畴中的归纳和限制函子，推广了可分扩张的一些经典结果。对$A ^σ _α G$的整体维数和有限维数进行了分类：$A ^σ _α G$的整体维数为无穷大或等于$A$的整体维数，$A ^σ _α G$的有限维数与$A$的有限维数一致.给出了斜群环具有有限整体维数的一个判别准则。在$A$是一个半准素代数的假设下，证明了$p$和$S \leqslant G$在额外的假设下具有相同的同调维数，推广了作者在以前的一些文章中的主要结果.
英文摘要：
In this paper we consider several homological dimensions of crossed products $A _α ^σ G$, where $A$ is a left Noetherian ring and $G$ is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of $A ^σ _α G$ are classified: global dimension of $A ^σ _α G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^σ _α G$ coincides with that of $A$. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that $A$ is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow $p$-subgroup $S \leqslant G$, we show that $A$ and $A _α ^σ G$ share the same homological dimensions under extra assumptions, extending the main results of the author in some previous papers.
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