Enumerating Non-Stable Vector Bundles

Abstract

In this article, we establish a motivic analog of an enumeration result of James-Thomas on non-stable vector bundles in topological setting. Using this, we obtain results on enumeration of projective modules of rank $d$ over a smooth affine $k$-algebra $A$ of dimension $d$, recovering in particular results of Suslin and Bhatwadekar on cancellation of such vector bundles. Admitting a conjecture of Asok and Fasel, we prove cancellation of such vector bundles of rank $d-1$ if the base field $k$ is algebraically closed. We also explore the cancellation properties of symplectic vector bundles.

arXiv commentNew version, 37 pages. Added a cancellation result for symplectic vector bundles and exposition improved. Comments are welcome!

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• Algebraic vector bundles and $p$-local A^1-homotopy theory