Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the Chevalley–Warning problem in Grothendieck ring

Le Dang Thi Nguyen

doi:10.1016/j.crma.2012.05.008

Algebraic Geometry

Unramified cohomology,

A^{1}

-connectedness, and the Chevalley–Warning problem in Grothendieck ring
[Cohomologie non ramifiée,

A^{1}

-connexité et le problème de Chevalley–Warning dans lʼanneau de Grothendieck]

Présenté par : Claire Voisin

Le Dang Thi Nguyen ¹

¹ Mathematik, Universität Duisburg–Essen, Universitätstr., 45117 Essen, Germany

Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 613-615.

Résumé
Abstract

Nous étudions le problème de Chevalley–Warning dans lʼanneau de Grothendieck $K_{0} (Var / k)$ . Nous montrons que la théorie $A^{1}$ -homotopie fournit des invariants sur $K_{0} (Var / k) / L$ . 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 $C_{1}$ (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 $K_{0} (Var / k)$ . We show that the $A^{1}$ -homotopy theory yields well-defined invariants on $K_{0} (Var / k) / L$ , 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 $C_{1}$ -field (Brown and Schnetz, 2011 [6]). This also gives a negative answer to the question in Bilgin (2011) [5, Ques. 3.8].

Reçu le : 2012-02-29
Accepté le : 2012-05-21
Publié le : 2012-06-01

DOI : 10.1016/j.crma.2012.05.008

Affiliations des auteurs :

Le Dang Thi Nguyen ¹

¹ Mathematik, Universität Duisburg–Essen, Universitätstr., 45117 Essen, Germany

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     title = {Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the {Chevalley{\textendash}Warning} problem in {Grothendieck} ring},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {613--615},
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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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