Variations in ${\mathbb{A}}^1$ on a theme of Mohan Kumar

Abstract

For every prime $p$, Mohan Kumar constructed examples of stably free modules of rank $p$ on suitable $(p+1)$-dimensional smooth affine varieties. This note discusses how to detect the corresponding unimodular rows in motivic cohomology. Using the recent developments in the ${\mathbb{A}}^1$-obstruction classification of vector bundles, this provides an alternative proof of non-triviality of Mohan Kumar's stably free modules. The reinterpretation of Mohan Kumar's examples also allows to produce interesting examples of stably trivial torsors for other algebraic groups.

23 pages, comments welcome! Revision includes full details for the computation of the cohomology operation using the twisted Gersten complexes and fixes a 2-torsion mistake in a previous version

References in corpus (2)

• From algebraic cobordism to motivic cohomology
• Euler class groups and motivic stable cohomotopy