A remark on the Farrell-Jones conjecture

TL;DR

This paper explores the Farrell-Jones conjecture, constructing a specific example where the associated Waldhausen K-theory space simplifies to a rational Eilenberg-MacLane space, highlighting implications for algebraic K-theory.

Contribution

It provides an explicit example under the Farrell-Jones conjecture where the K-theory space is a rational Eilenberg-MacLane space, offering new insights into the conjecture's consequences.

Findings

Constructed a specific group ring R and subcategory C

Demonstrated K(C) is equivalent to a rational Eilenberg-MacLane space

Highlights implications for algebraic K-theory and the Farrell-Jones conjecture

Abstract

Assuming the classical Farrell-Jones conjecture we produce an explicit (commutative) group ring $R$ and a thick subcategory $\mathsf{C}$ of perfect $R$-complexes such that the Waldhausen $K$-theory space $\mathrm{K}(\mathsf{C})$ is equivalent to a rational Eilenberg-Maclane space.