Homological dimensions of crossed products

Liping Li

K-Theory and Homology Representation Theory Rings and Algebras mathscidoc:1702.20002

to appear in Glasgow Math. J., 2017
In this paper we consider several homological dimensions of crossed products $A _{\alpha} ^{\sigma} G$, 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 $A ^{\sigma} _{\alpha} G$ are classified: global dimension of $A ^{\sigma} _{\alpha} G$ is either infinity or equal to that of $A$, and finitistic dimension of $A ^{\sigma} _{\alpha} G$ 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 $A _{\alpha} ^{\sigma} G$ share the same homological dimensions under extra assumptions.
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  title={Homological dimensions of crossed products},
  author={Liping Li},
  booktitle={to appear in Glasgow Math. J.},
Liping Li. Homological dimensions of crossed products. 2017. In to appear in Glasgow Math. J.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170222105618445933501.
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