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arXiv · 2110.01413

The passage from the integral to the rational group ring in algebraic $K$-theory

Georg Lehner
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Abstract

An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to the rational group ring is trivial for any group $G$. We will show that this is false, with a counterexample given by the group $QD_{32} *_{Q_{16}} QD_{32}$. We will also show how to compute the image of the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ using representation theoretic means, assuming $G$ satisfies the Farrell-Jones conjecture.

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Georg Lehner. 2021-10-04. The passage from the integral to the rational group ring in algebraic $K$-theory. https://arxiv.org/abs/2110.01413

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