Secondary characteristic classes and the Euler class. (English) Zbl 1349.14078

Asok, Aravind; Fasel, Jean

Doc. Math. Extra Vol., Alexander S. Merkurjev’s Sixtieth Birthday, 7-29 (2015).

Summary: We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if \(X\) is a smooth affine scheme of dimension \(d\) over a field \(k\) of finite 2-cohomological dimension (with \(\mathrm{char}(k)\neq 2)\) 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.

MSC:

14F42	Motivic cohomology; motivic homotopy theory
14C15	(Equivariant) Chow groups and rings; motives
13C10	Projective and free modules and ideals in commutative rings
55S20	Secondary and higher cohomology operations in algebraic topology