Groups of permutation projective dimension two

Author:
James E. Arnold

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

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Abstract: In previous papers we developed a version of homological algebra for modules ( 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. -torsion free module is either projective or has infinite projective dimension, whereas for cyclic, all f.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.

**[1]**J. E. Arnold, Jr.,*A generalized Cartan isomorphism for the Grothendieck group of a finite group*, J. Pure Appl. Algebra**12**(1978), 225-234. MR**501950 (80a:18012)****[2]**-,*Homological algebra based on permutation modules*, J. Algebra**70**(1981), 250-260. MR**618392 (83k:20056)****[3]**H. Cartan and S. Eilenberg,*Homological algebra*, Princeton Univ. Press, Princeton, N. J., 1956. MR**0077480 (17:1040e)****[4]**S. Endo and T. Miyata,*On a classification of the function fields of algebraic tori*, Nagoya Math. J.**56**(1975), 85-104. MR**0364203 (51:458)****[5]**-,*Integral representations with trivial first cohomology groups*, preprint.

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DOI:
https://doi.org/10.1090/S0002-9939-1984-0746077-6

Article copyright:
© Copyright 1984
American Mathematical Society