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Talk

January 18, 2022
Georg Lehner (FU Berlin): The passage from the integral to the rational group ring in algebraic K-theory

Abstract

Let G be a group, ZG and QG be the integral and rational group ring. A known theorem due to Swan states that, if P is finitely generated, projective ZG-module, then its rationalization P \otimes Q is free as a QG-module. There have been many attempts to generalize Swans theorem to infinite groups, notably the Bass trace conjectures. Lück, Reich formulated what is known as the integral K₀(ZG)-to-K₀(QG)-conjecture, which states that the map induced by rationalization on the algebraic K-theory groups from K₀(ZG) to K₀(QG) has image only in the subgroup of K₀(QG) generated by the free modules. We will show that the integral K₀(ZG)-to-K₀(QG)-conjecture is false and construct a concrete counterexample as an amalgamated product G =K*_H K, where K and H are suitable groups allowing particular quaternion representations. We also show that the Farrell-Jones conjecture for G implies that the image of K₀(ZG) to K₀(QG) away from the free modules is in fact 2-torsion and give a geometric vanishing criterion in terms of the structure of the classifying space E(G;Fin) of the family of finite subgroups of G.

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