The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture

We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in $RG$ are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.