Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T16:05:34.716Z Has data issue: false hasContentIssue false
  • English
  • Français

FINITISTIC DIMENSIONS AND PIECEWISE HEREDITARY PROPERTY OF SKEW GROUP ALGEBRAS

Part of: Homological methods Representation theory of rings and algebras

Published online by Cambridge University Press:  18 December 2014

LIPING LI
LIPING LI*
Affiliation:
Department of Mathematics, University of California, Riverside, CA, USA, 92521 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 16E10: Homological dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 509 - 517
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Auslander, M., Reiten, I. and Smalø, S., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36 (Cambridge University Press, Cambridge, UK, 1997).Google Scholar
2.Boisen, P., The representation theory of full group-graded algebras, J. Algebra 151 (1) (1992), 160–179.CrossRefGoogle Scholar
3.Cohen, M. and Montgomery, S., Group-graded rings, smash products, and group actions, Trans. Am. Math. Soc. 282 (1) (1984), 237–258.CrossRefGoogle Scholar
4.Dionne, J., Lanzilotta, M. and Smith, D., Skew group algebras of piecewise hereditary algebras are piecewise hereditary, J. Pure Appl. Algebra 213 (2) (2009), 241–249.CrossRefGoogle Scholar
5.Happel, D., Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119 (Cambridge University Press, Cambridge, UK, 1988).CrossRefGoogle Scholar
6.Happel, D. and Zacharia, D., A homological characterization of piecewise hereditary algebras, Math. Z. 260 (1) (2008), 177–185.CrossRefGoogle Scholar
7.Happel, D. and Zacharia, D., Homological properties of piecewise hereditary algebras, J. Algebra 323 (4) (2010), 1139–1154.CrossRefGoogle Scholar
8.Happel, D., Reiten, I. and Smalø, S., Piecesise hereditary algebras, Arch. Math. 66 (3) (1996), 182–186.CrossRefGoogle Scholar
9.Li, L., Representations of modular skew group algebras, Trans. Am. Math. Soc., to appear in Trans. Am. Math. Soc.Google Scholar
10.Marcus, A., Representation theory of group graded algebras, (Nova Science Publishers, Inc., Commack, NY, USA, 1999).Google Scholar
11.Martinez, R., Skew group algebras and their Yoneda algebras Math. J. Okayama Univ. 43 (2001), 1–16.Google Scholar
12.Passman, D., Group rings, crossed products and Galois theory, CBMS Regional Conference Series in Mathematics (64), 1986.CrossRefGoogle Scholar
13.Reiten, I. and Riedtmann, C., Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1) (1985), 224–282.CrossRefGoogle Scholar
14.Zhong, Y., Homological dimension of skew group rings and cross product, J. Algebra 164 (1) (1994), 101–123.Google Scholar
15.Zimmermann, B., The finitistic dimension conjectures - a tale of 3.5 decades, Abelian groups and modules (Padova, 1994), Math. Appl., vol. 343 (Kluwer Acad. Publ., Dordrecht, 1995), 501–517.Google Scholar