Efficient generation of projective modules: a motivic view

Assume $k$ is a field and $R$ is a smooth $k$-algebra of dimension $d$. If $P$ is a projective module of rank $r$, then it is well-known that $P$ can be generated by $r+d$-elements (Forster--Swan). Under suitable assumptions on $r$ and $d$, we investigate obstructions to generation of $P$ by fewer than $r+d$ elements using motivic homotopy theory. For example, we observe that a quadratic enhancement of the classical Segre class obstructs generation by $r+d-1$ elements, whether or not $k$ is algebraically closed, generalizing old results of M.P. Murthy. Along the way, we also establish efficient generation results for symplectic modules.