Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-11T13:14:04.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank Element of a Projective Module

Published online by Cambridge University Press:  22 January 2016

Akira Hattori
Akira Hattori*
Affiliation:
Tokyo University of Education
Rights & Permissions [Opens in a new window]

Extract

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.

In § 1 of this note we first define the trace of an endomorphism of a projective module P over a non-commutative ring A. Then we call the trace of the identity the rank element r(P) of P, which we shall illustrate by several examples. For a projective module P over the groupalgebra of a finite group G, the rank element of P is essentially the character of G in P. In § 2 we prove that under certain assumption two projective modules Pi and P2 over an algebra over a complete local ring o are isomorphic if and only if their rank elements are identical. This is a type of proposition asserting that two representations are equivalent if and only if their characters are identical, and in fact, when A is the groupalgebra, the above theorem may be considered as another formulation of Swan’s local theorem [9]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 113 - 120
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Auslander, M. and Goldman, O., The Brauer group of a commutative ring, Trans. Amer. Math. Soc., 97 (1960), 367–367.CrossRefGoogle Scholar
[2] Bass, H., Projective modules over algebras, Ann. of Math., 73 (1961), 532–532.CrossRefGoogle Scholar
[3] Curtis, C. and Reiner, I., Representation theory of finite groups and associative algebras, Interscience, 1962.Google Scholar
[4] Giorgiutti, I., Modules project if s sur les algebres de groupes finis, C. R. Paris, 250 (1960), 1419–1420.Google Scholar
[5] Hattori, A., Integral representations and projective modules, Proceedings of the Symposium on Group Theory, Tokyo, (1961), 95–95 (in Japanese).Google Scholar
[6] Hattori, A., Semisimple algebras over a commutative ring, J. Math. Soc. Japan, 15 (1963), 404–404.CrossRefGoogle Scholar
[7] Nakayama, T., On modules of trivial cohomology over a finite group II, Nagoya Math. J., 12 (1957), 171–171.CrossRefGoogle Scholar
[8] Serre, J.-P., Corps locaux, Hermann, 1962.Google Scholar
[9] Swan, R., Induced representations and projective modules, Ann. of Math., 71 (1960),552–552.Google Scholar