Comptes Rendus
no. 11-12
Algebraic Geometry
Unramified cohomology, A1-connectedness, and the Chevalley–Warning problem in Grothendieck ring
[Cohomologie non ramifiée, A1-connexité et le problème de Chevalley–Warning dans lʼanneau de Grothendieck]

Présenté par : Claire Voisin

Le Dang Thi Nguyen1
1 Mathematik, Universität Duisburg–Essen, Universitätstr., 45117 Essen, Germany
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 613-615.

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].

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].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.05.008
Le Dang Thi Nguyen 1

1 Mathematik, Universität Duisburg–Essen, Universitätstr., 45117 Essen, Germany 
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Le Dang Thi Nguyen. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.008/

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Cité par Sources :

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

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