Cancellation of vector bundles of rank $3$ with trivial Chern classes on smooth affine fourfolds
Tariq Syed

TL;DR
This paper proves that rank 3 projective modules with trivial Chern classes over certain smooth affine fourfolds are cancellative, extending previous results and employing a generalized Vaserstein symbol sum formula.
Contribution
It introduces a sum formula for the generalized Vaserstein symbol applicable to smooth affine algebras over perfect fields, leading to new cancellation results for rank 3 modules.
Findings
Proves a sum formula for the generalized Vaserstein symbol when n ≡ 0,1 mod 4.
Extends Fasel-Rao-Swan results on unimodular row transformations.
Shows all rank 3 projective modules with trivial Chern classes over certain fourfolds are cancellative.
Abstract
If , we prove a sum formula for the generalized Vaserstein symbol whenever is a smooth affine algebra over a perfect field with such that . This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension over algebraically closed fields of characteristic . As a consequence, we prove that any projective module of rank with trivial Chern classes over a smooth affine algebra of dimension over an algebraically closed field with is cancellative.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Algebra and Geometry
