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

TL;DR

This paper explores the detection of certain algebraic modules in motivic cohomology, providing an alternative proof of their non-triviality and extending the construction to other algebraic groups using recent $ ext{A}^1$-obstruction classification methods.

Contribution

It offers a new approach to verify the non-triviality of Mohan Kumar's stably free modules via motivic cohomology and extends the examples to other algebraic groups.

Findings

Alternative proof of non-triviality of Mohan Kumar's modules

Detection of unimodular rows in motivic cohomology

Construction of stably trivial torsors for other algebraic groups

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.