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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Localized group rings, the invariant basis property and Euler characteristics


Author: K. R. Goodearl
Journal: Proc. Amer. Math. Soc. 91 (1984), 523-528
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 $ {W^ * }$ group algebra is exploited to prove that certain localizations of a group ring $ KG$ possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain $ KG$-modules have zero Euler characteristic. The assumptions are that $ K$ is a commutative integral domain of characteristic zero, and that $ G$ has a nontrivial, torsion-free, abelian normal subgroup $ A$. The main result of the paper is that the localization of $ KG$ obtained by inverting all elements of the form $ \alpha - a$, where $ \alpha $ is a nonzero element of $ K$ and $ a$ is a nontrivial element of $ A$, 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 $ KG$ obtained by inverting all nonzero elements of $ KA$.) Given a $ KG$-module $ M$ which has a finite free resolution, such that $ M$ is finitely generated over $ K$, it is proved that the Euler characteristic of $ M$ is zero. This verifies an unpublished result of Rosset.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0746081-8
PII: S 0002-9939(1984)0746081-8
Keywords: Group ring, invariant basis property, direct finiteness, Euler characteristic
Article copyright: © Copyright 1984 American Mathematical Society