The cancellation of projective modules of rank 2 with a trivial determinant

Syed, Tariq

doi:10.2140/ant.2021.15.109

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The cancellation of projective modules of rank 2 with a trivial determinant

Tariq Syed

Vol. 15 (2021), No. 1, 109–140

DOI: 10.2140/ant.2021.15.109

Abstract

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\leq 4$ . If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6 \in k^{\times}$ , then we prove that stably free $R$ -modules of rank $2$ are free if and only if a Hermitian $K$ -theory group ${\tilde{V}}_{SL} (R)$ is trivial.

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Keywords

cancellation, projective module, stably free module, Vaserstein symbol

Mathematical Subject Classification 2010

Primary: 19A13

Secondary: 13C10, 14F42, 19G38

Milestones

Received: 16 July 2019

Revised: 3 June 2020

Accepted: 2 July 2020

Published: 1 March 2021

Authors

Tariq Syed
Fakultät für Mathematik Universität Duisburg-Essen Essen Germany

Vol. 15, No. 1, 2021