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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 

References

REFERENCES

1. Aljadeff, E., On the surjectivity of some trace maps, Israel J. Math. 86 (1–3) (1994), 221–232.Google Scholar
2. Aljadeff, E. and Ginosar, Y., Induction from elementary abelian subgroups, J. Algebra 179 (2) (1996), 599–606.Google Scholar
3. Auslander, M., Reiten, I. and Smalø, S., Representation theory of Artin algebras, Cambridge Studies in Advance Mathematics, vol. 36 (Cambridge University Press, 1997).Google Scholar
4. Avramov, L. and Foxby, H., Homological dimensions of unbounded complexes, J. Pure App. Algebra 71 (2–3) (1991), 129–155.CrossRefGoogle Scholar
5. Benson, D. J., Representations and cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advance Mathematics, vol. 30 (Cambridge University Press, 1991).Google Scholar
6. Boisen, P., The representation theory of full group-graded algebras, J. Algebra 151 (1) (1992), 160–179.Google Scholar
7. Cohen, M. and Montgomery, S., Group-graded rings, smash products, and group actions, Tran. Amer. Math. Soc. 282 (1) (1984), 237–258.Google Scholar
8. Conlon, S., Twisted group algebras and their representations, J. Austral. Math. Soc. 4 (1964), 152–173.Google Scholar
9. Happel, D. and Zacharia, D., A homological characterization of piecewise hereditary algebras, Math. Z. 260 (1) (2008), 177–185.CrossRefGoogle Scholar
10. Happel, D. and Zacharia, D., Homological properties of piecewise hereditary algebras, J. Algebra 323 (4) (2010), 1139–1154.Google Scholar
11. Hirata, K. and Sugano, K., On semisimple extensions and separable extensions over non commutative rings, J. Math. Soc. Japan 18 (1966), 360–373.Google Scholar
12. Iacob, A. and Iyengar, S., Homological dimensions and regular rings, J. Algebra 322 (10) (2009), 3451–3458.CrossRefGoogle Scholar
13. Igusa, K., Liu, S. and Paquette, C., A proof of the strong no loop conjecture, Adv. Math. 228 (5) (2011), 2731–2742.Google Scholar
14. Karpilovsky, G., The algebraic structure of crossed products, North-Holland Mathematical Studies, vol. 142 (North-Holland Publishing Co., Amsterdam, 1987). x+348 pp.Google Scholar
15. Li, L., Representations of modular skew group algebras, Trans. Amer. Math. Soc. 367 (9) (2015), 6293–6314.Google Scholar
16. Li, L., Finitistic dimensions and picewise hereditary property of skew group algebras, Glasgow Math. J. 57 (3) (2015), 509–517.CrossRefGoogle Scholar
17. Lundström, P., Crossed product algebras defined by separable extensions, J. Algebra 283 (2) (2005), 723–737.CrossRefGoogle Scholar
18. Marcus, A., Representation theory of group graded algebras (Nova Science Publishers Inc., Commack, NY, 1999).Google Scholar
19. Montgomery, S. and Witherspoon, S., Irreducible representations of crossed products, J. Pure App. Algebra 129 (3) (1998), 315–326.Google Scholar
20. Nǎstǎasescu, C., Van den Bergh, M. and Van Oystaeyen, F., Separable functors applied to graded rings, J. Algebra 123 (2) (1989), 397–413.Google Scholar
21. Passman, D., Group rings, crossed products and Galois theory, CBMS Regional Conference Series in Mathematics, vol. 64 (American Mathematical Society, Providence, RI, 1986). viii+71 pp.Google Scholar
22. Passman, D., Infinite crossed products, Pure and Applied Mathematics, vol. 135 (Academic Press, Inc., Boston, MA, 1989). xii+468 pp.Google Scholar
23. Reiten, I. and Riedtmann, C., Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1) (1985), 224–282.Google Scholar
24. Yi, Z., Homological dimension of skew group rings and cross product, J. Algebra 164 (1) (1994), 101–123.Google Scholar