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HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS

Part of: Homological methods Homological algebra

Published online by Cambridge University Press:  10 June 2016

LIPING LI
LIPING LI*
Affiliation:
College of Mathematics and Computer Science, Performance Computing and Stochastic Information Processing (Ministry of Education), Hunan Normal University, Changsha, Hunan 410081, China e-mail: lipingli@hunnu.edu.cn
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Abstract

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In this paper, we consider several homological dimensions of crossed products AασG, 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 AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG 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 ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

MSC classification

Primary: 18G20: Homological dimension
Secondary: 16E10: Homological dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 401 - 420
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

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