Free summands of stably free modules

Abstract Let R be a commutative ring. One may ask when a general R -module P that satisfies $P \oplus R \cong R^n$ has a free summand of a given rank. M. Raynaud translated this question into one about sections of certain maps between Stiefel varieties: if $V_r(\mathbb {A}^n)$ denotes the variety $\operatorname {GL}(n) / \operatorname {GL}(n-r)$ over a field k , then the projection $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section if and only if the following holds: any module P over any k -algebra R with the property that $P \oplus R \cong R^n$ has a free summand of rank $r-1$ . Using techniques from $\mathbb {A}^1$ -homotopy theory, we characterize those n for which the map $V_r(\mathbb {A}^n) \to V_1(\mathbb {A}^n)$ has a section in the cases $r=3,4$ under some assumptions on the base field. We conclude that if $P \oplus R \cong R^{24m}$ and R contains the field of rational numbers, then P contains a free summand of rank $2$ . If R contains a quadratically closed field of characteristic $0$ , or the field of real numbers, then P contains a free summand of rank $3$ . The analogous results hold for schemes and vector bundles over them.