Let $R$ be a commutative ring containing an algebraically closed field of characteristic $0$. An $R$-module $P$ is said to be stably free of type $(n,n-1)$ if there is an isomorphism $P \oplus R = R^n$ For what values of $n$ and $q$ do stably free modules of type $(n,n-1)$ necessarily admit free summands of rank $q-1$? This question was important in the history of applying topology to algebra. In 1968, Raynaud showed that the James number $b_q$ must divide $n$, using comparison of étale and singular cohomology.

By converting the problem to one in motivic homotopy theory, and then relating it to comparison results for the homotopy groups of the sphere spectrum, we show that Raynaud's necessary condition is actually sufficient. This is joint work with Sebastian Gant.