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 Stiefel variety $\text{GL}(n)/\text{GL}(n-r)$ 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 a field of characteristic $0$, 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.

15 pages