Free summands of stably free modules

TL;DR

This paper uses $ ext{A}^1$-homotopy theory to determine when stably free modules over a ring contain free summands of specific ranks, providing new criteria based on the base field and module rank.

Contribution

It characterizes the existence of free summands in stably free modules via $ ext{A}^1$-homotopy theory, especially for ranks 3 and 4, under certain field assumptions.

Findings

Modules over rings with $P igoplus R o R^{24m}$ contain free summands of rank 2 if $R$ has characteristic 0.

Modules over rings with a quadratically closed field or real numbers contain free summands of rank 3.

Results extend to schemes and vector bundles, broadening classical algebraic $K$-theory insights.

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 $\textrm{GL}(n) / \textrm{GL}(n-r)$ 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.