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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
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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Additional Information

Hélène Esnault
Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
Email: esnault@uni-essen.de

Bruno Kahn
Affiliation: Institut de Mathématiques de Jussieu, Université Paris 7, Case 7012, 75251 Paris Cedex 05, France
Email: kahn@math.jussieu.fr

Marc Levine
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: marc@neu.edu

Eckart Viehweg
Affiliation: FB6, Mathematik, Universität Essen, D-45117 Essen, Germany
Email: viehweg@uni-essen.de

DOI: http://dx.doi.org/10.1090/S0894-0347-98-00248-3
PII: S 0894-0347(98)00248-3
Received by editor(s): September 13, 1996
Received by editor(s) in revised form: July 28, 1997
Additional Notes: This research was partially supported by the DFG Forschergruppe “Arithmetik und Geometrie”; the second and third author gratefully acknowledge its hospitality.
Article copyright: © Copyright 1998 American Mathematical Society