The Arason invariant and mod 2 algebraic cycles, by Helene Esnault, Bruno Kahn, Marc Levine, and Eckart Viehweg

Let X be a smooth, irreducible variety over a field of characteristic different from 2, E a quadratic bundle over X and q its generic fiber, which is a quadratic form over the function field K of X. Assume that q belongs to I^3K, so that its Arason invariant e, a degree 3 Galois cohomology class with coefficients Z/2, is defined. This class is unramified over X. We give a formula for d(e) in CH^2(X)/2, where CH^2(X) is the second Chow group of X and d is a Bloch-Ogus differential:

d(e)= c_2(E) +c(E)^2

where c_2(E) is the (cycle-theoretic) second Chern class of the underlying vector bundle to E and c(E) is the Clifford invariant of E, which turns out to be a codimension 1 cycle modulo 2.

As an application we get the following theorem. Let k be a field of characteristic different from 2, q a quadratic form over k which belongs to I^2k, C(q) the Clifford algebra of q and K the function field of the Severi-Brauer variety associated to C(q), so that q_K is in I^3K. Assume that the index of C(q) is at least 8. Then the Arason invariant of q_K is nonzero, so that q_K is not in I^4K. In particular, if dim q = 8, then q_K is anisotropic.

The method to prove the formula involves computing low-degree K-cohomology of certain reductive groups and (of) their simplicial classifying schemes.


Helene Esnault <esnault@uni-essen.de>
Bruno Kahn <kahn@mathp7.jussieu.fr>
Marc Levine <marc@neu.edu>
Eckart Viehweg <viehweg@uni-essen.de>