A cohomological classification of vector bundles on smooth affine threefolds

Details A cohomological classification of vector bundles on smooth affine threefolds A cohomological classification of vector bundles on smooth affine threefolds

2012-04-03

arxiv.org/abs/1204.0770v2

Mathematics

Algebraic Geometry

49 pages; Some parts moved to http://arxiv.org/abs/1204.4538, Comments still welcome!

Scientific paper

We give a cohomological classification of vector bundles on smooth affine threefolds over algebraically closed fields having characteristic unequal to 2. As a consequence, if A is a smooth affine algebra of dimension 3 over an algebraically closed field having characteristic unequal to 2, we deduce that cancellation holds for arbitrary rank projective modules. The proofs of these results involve three main ingredients. First, we give a description of the second unstable A^1-homotopy sheaf of the general linear group. Second, these computations can be used in concert with F. Morel's A^1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A^1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.

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