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Determinants in Projective Modules1)

Published online by Cambridge University Press:  22 January 2016

Oscar Goldman
Oscar Goldman*
Affiliation:
Brandeis University
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The definition of the determinant of an endomorphism of a free module depends on the following fact: If F is a free R-module of rank n, then the homogeneous component ∧nF, of degree n, of the exterior algebra ∧ F of F is a free R-module of rank one. If a is an endomorphism of F, then a extends to an endomorphism of ∧ F which in ∧nF is therefore multiplication by an element of R. That factor is then defined to be the determinant of α. (A discussion of this theory may be found in [4].)

This procedure cannot be applied in general to finitely generated projective modules since, for such modules, it may happen that no homogeneous component of the exterior algebra is free of rank one.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 18 , April 1961 , pp. 27 - 36
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1961

Footnotes

1)

This work was performed under a grant from the National Science Foundation.

References

[1] Auslander, M. and Goldman, O., Maximal Orders, Trans. Amer. Math. Soc., 97(1960), pp. 1–24.CrossRefGoogle Scholar
[2] Birkhoff, G. and MacLane, S., A survey of modern algebra, New York, 1953.Google Scholar
[3] Cartan, H. and Eilenberg, S., Homological algebra, Princeton, 1956.Google Scholar
[4] Chevalley, C., Fundamental concepts of algebra, New York, 1956.Google Scholar