Secondary characteristic classes and the Euler class
Secondary characteristic classes and the Euler class
We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We show that if X is a smooth affine scheme of dimension d over a field k of finite 2-cohomological dimension (with char(k) $\neq$ 2) and E is a rank d vector bundle over X, vanishing of the Chow-Witt theoretic Euler class of E is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when k is of small 2-cohomological dimension.
21 pages; Final version to appear in Doc. Math. (differs in content rather significantly from v1)
- Grenoble Alpes University France
- French National Centre for Scientific Research France
- Institut Fourier France
- University of Southern California United States
- University of California System United States
K-Theory and Homology (math.KT), Projective and free modules and ideals in commutative rings, [MATH] Mathematics [math], Mathematics - Commutative Algebra, Commutative Algebra (math.AC), 14F42, 14C15, 13C10, 55S20, Mathematics - Algebraic Geometry, Motivic cohomology; motivic homotopy theory, (Equivariant) Chow groups and rings; motives, Secondary and higher cohomology operations in algebraic topology, Mathematics - K-Theory and Homology, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Algebraic Geometry (math.AG)
K-Theory and Homology (math.KT), Projective and free modules and ideals in commutative rings, [MATH] Mathematics [math], Mathematics - Commutative Algebra, Commutative Algebra (math.AC), 14F42, 14C15, 13C10, 55S20, Mathematics - Algebraic Geometry, Motivic cohomology; motivic homotopy theory, (Equivariant) Chow groups and rings; motives, Secondary and higher cohomology operations in algebraic topology, Mathematics - K-Theory and Homology, FOS: Mathematics, Algebraic Topology (math.AT), Mathematics - Algebraic Topology, Algebraic Geometry (math.AG)
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