Efficient generation of projective modules: a motivic view
Aravind Asok, Morgan Opie, Brian Shin, Tariq Syed
TL;DR
The paper develops motivic obstruction theory to study when a rank $r$ projective module over a smooth affine $k$-algebra of dimension $d$ can be generated by fewer than $r+d$ elements. It identifies the primary obstruction to $r+d-1$-generated modules with a quadratically enhanced Segre class and, under various hypotheses on $k$ and $X$, characterizes when $r+d-1$ or even $r+d-2$ generators suffice via Segre/Euler classes and twisted Chow–Witt groups. The framework extends Murthy’s results to non-algebraically closed bases and integrates obstruction theory with explicit computations of $\\pi_i$-sheaves of Stiefel varieties, including stable ranges and exact sequences, to produce concrete generation criteria. A symplectic analogue is established, yielding a Forster–Swan-type theorem for symplectic modules and clarifying when such modules embed into lower-rank hyperbolic symplectic modules. Collectively, the work provides a motivic, obstruction-theoretic pipeline linking Grassmannian lifts, characteristic classes, and Chow–Witt data to practical generation bounds for projective and symplectic bundles over smooth affine varieties.
Abstract
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.