5 problems

Let $R$ be a regular ring and let $G$ be a group. The negative K-theory conjecture. If $G$ is torsionfree, then … This is the negative-degree part of the Farrell–Jones program for…

Let $G$ be a group and let $\operatorname{Wh}(G)$ denote its Whitehead group. The Whitehead group conjecture. If $G$ is torsionfree, then $\operatorname{Wh}(G)$ vanishes. This woul…

Let $R$ be a field and let $G$ be a group. Kaplansky's idempotent conjecture. If $G$ is torsionfree, then the group ring $RG$ has only the trivial idempotents, namely $0$ and $1$.…

Let $G$ be a group and let $R$ be a regular ring, meaning that $R$ is Noetherian and every $R$-module has a finite-dimensional projective resolution. The regular-ring projective cl…

Let $G$ be a group and let $\widetilde{K}_0(\mathbb{Z}G)$ denote its reduced projective class group. The reduced projective class group conjecture. The group…