Groups of permutation projective dimension two
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- by James E. Arnold
- Proc. Amer. Math. Soc. 91 (1984), 505-509
- DOI: https://doi.org/10.1090/S0002-9939-1984-0746077-6
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Abstract:
In previous papers we developed a version of homological algebra for $Z[G]$ modules ($G$ a finite group) using summands of permutation modules in place of projective modules. The resulting theory is more discriminating than the usual homological algebra. For example, an f.g. $Z$-torsion free $Z[G]$ module is either projective or has infinite projective dimension, whereas for $G$ cyclic, all f.g. $Z[G]$ modules have permutation projective dimension one. In this paper we apply results of Endo and Miyata on permutation projective modules to characterize groups of dimension two.References
- James E. Arnold Jr., A generalized Cartan isomorphism for the Grothendieck group of a finite group, J. Pure Appl. Algebra 12 (1978), no. 3, 225–234. MR 501950, DOI 10.1016/0022-4049(87)90003-X
- James E. Arnold Jr., Homological algebra based on permutation modules, J. Algebra 70 (1981), no. 1, 250–260. MR 618392, DOI 10.1016/0021-8693(81)90257-X
- Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
- Shizuo Endô and Takehiko Miyata, On a classification of the function fields of algebraic tori, Nagoya Math. J. 56 (1975), 85–104. MR 364203 —, Integral representations with trivial first cohomology groups, preprint.
Bibliographic Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 505-509
- MSC: Primary 20J06; Secondary 18G20
- DOI: https://doi.org/10.1090/S0002-9939-1984-0746077-6
- MathSciNet review: 746077