Localized group rings, the invariant basis property and Euler characteristics

Goodearl, K. R.

doi:10.1090/S0002-9939-1984-0746081-8

Localized group rings, the invariant basis property and Euler characteristics
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Proc. Amer. Math. Soc. 91 (1984), 523-528 Request permission

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 This verifies an unpublished result of Rosset.

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Lectures on Von Neumann algebras