On the image of the trivial source ring in the ring of virtual characters of a finite group

We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Co\c{s}kun’s theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc’s theory of rational $p$-biset functors.