Determinants in Projective Modules
- 1 April 1961
- journal article
- research article
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- Vol. 18, 27-36
- https://doi.org/10.1017/s002776300000221x
Abstract
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.Keywords
This publication has 1 reference indexed in Scilit:
- Maximal ordersTransactions of the American Mathematical Society, 1960