Algebraic Geometry
Unramified cohomology, A1-connectedness, and the Chevalley–Warning problem in Grothendieck ring
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 613-615.

We study the Chevalley–Warning problem in the Grothendieck ring K0(Var/k). We show that the A1-homotopy theory yields well-defined invariants on K0(Var/k)/L, in particular the Brauer group is such an invariant. We use this to give a concrete counter-example to the Chevalley–Warning conjecture over a C1-field (Brown and Schnetz, 2011 [6]). This also gives a negative answer to the question in Bilgin (2011) [5, Ques. 3.8].

Nous étudions le problème de Chevalley–Warning dans lʼanneau de Grothendieck K0(Var/k). Nous montrons que la théorie A1-homotopie fournit des invariants sur K0(Var/k)/L. En particulier le groupe de Brauer est un tel invariant. Nous utilisons cela pour donner un contre-exemple concret à la conjecture de Chevalley–Warning sur un corps C1 (Brown et Schnetz, 2011 [6]). Cela donne aussi une réponse négative à la question dans Belgin (2011) [5, Ques. 3.8].

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DOI: 10.1016/j.crma.2012.05.008
Nguyen, Le Dang Thi 1

1 Mathematik, Universität Duisburg–Essen, Universitätstr., 45117 Essen, Germany
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Nguyen, Le Dang Thi. Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the Chevalley–Warning problem in Grothendieck ring. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 613-615. doi : 10.1016/j.crma.2012.05.008. http://www.numdam.org/articles/10.1016/j.crma.2012.05.008/

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This work has been supported by SFB/TR45 “Periods, moduli spaces and arithmetic of algebraic varieties”.