The Arason invariant and mod 2 algebraic cycles

Esnault, Hélène; Kahn, Bruno; Levine, Marc; Viehweg, Eckart

doi:10.1090/S0894-0347-98-00248-3

The Arason invariant and mod 2 algebraic cycles

Authors: Hélène Esnault, Bruno Kahn, Marc Levine and Eckart Viehweg
Journal: J. Amer. Math. Soc. 11 (1998), 73-118
MSC (1991): Primary 11E81; Secondary 55R40
DOI: https://doi.org/10.1090/S0894-0347-98-00248-3
MathSciNet review: 1460391
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Abstract: Let $k$ be a field, $X$ over $k$ a smooth variety with function field $K$ and $E$ a quadratic vector bundle over $X$ . Assuming that the generic fibre $q$ of $E$ is in $I^3K\subset W(K)$ , we compute the image of its Arason invariant

$\begin{displaymath}e^3(q)\in H^0(X,{\mathcal H}_{\mathrm{\acute{e}t}}^3({\mathbb Z}/2))\end{displaymath}$

in $CH^2(X)/2$ by the $d_2$ differential of the Bloch-Ogus spectral sequence. This gives an obstruction to $e^3(q)$ being a global cohomology class.

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