Localized group rings, the invariant basis property and Euler characteristics
Author:
K. R. Goodearl
Journal:
Proc. Amer. Math. Soc. 91 (1984), 523528
MSC:
Primary 16A27
MathSciNet review:
746081
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Abstract: The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding group algebra is exploited to prove that certain localizations of a group ring possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain modules have zero Euler characteristic. The assumptions are that is a commutative integral domain of characteristic zero, and that has a nontrivial, torsionfree, abelian normal subgroup . The main result of the paper is that the localization of obtained by inverting all elements of the form , where is a nonzero element of and is a nontrivial element of , has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of obtained by inverting all nonzero elements of .) Given a module which has a finite free resolution, such that is finitely generated over , it is proved that the Euler characteristic of is zero. This verifies an unpublished result of Rosset.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198407460818
PII:
S 00029939(1984)07460818
Keywords:
Group ring,
invariant basis property,
direct finiteness,
Euler characteristic
Article copyright:
© Copyright 1984
American Mathematical Society
