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

Nguyen, Le Dang Thi

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

Nguyen, Le Dang Thi ¹

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

Comptes Rendus. Mathématique, Tome 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 :

Nguyen, Le Dang Thi ¹

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

@article{CRMATH_2012__350_11-12_613_0,
     author = {Nguyen, Le Dang Thi},
     title = {Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the {Chevalley{\textendash}Warning} problem in {Grothendieck} ring},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {613--615},
     publisher = {Elsevier},
     volume = {350},
     number = {11-12},
     year = {2012},
     doi = {10.1016/j.crma.2012.05.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.05.008/}
}

TY  - JOUR
AU  - Nguyen, Le Dang Thi
TI  - Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the Chevalley–Warning problem in Grothendieck ring
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 613
EP  - 615
VL  - 350
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.05.008/
DO  - 10.1016/j.crma.2012.05.008
LA  - en
ID  - CRMATH_2012__350_11-12_613_0
ER  -

%0 Journal Article
%A Nguyen, Le Dang Thi
%T Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the Chevalley–Warning problem in Grothendieck ring
%J Comptes Rendus. Mathématique
%D 2012
%P 613-615
%V 350
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.05.008/
%R 10.1016/j.crma.2012.05.008
%G en
%F CRMATH_2012__350_11-12_613_0

Nguyen, Le Dang Thi. Unramified cohomology, $ {\mathbb{A}}^{1}$-connectedness, and the Chevalley–Warning problem in Grothendieck ring. Comptes Rendus. Mathématique, Tome 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/

Bibliographie
Cité par

[1] A. Asok, Birational invariants and $A^{1}$ -connectedness, Preprint, 2011, Crelleʼs J., , in press. | DOI

[2] Asok, A.; Morel, F. Smooth varieties up to $A^{1}$ -homotopy and algebraic h-cobordisms, Adv. Math., Volume 227 (2011) no. 5, pp. 1990-2058

[3] Ax, J. Zeroes of polynomials over finite fields, Amer. J. Math., Volume 86 (1964), pp. 255-261

[4] Beilinson, A.A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Luminy, 1981 (Astérisque), Volume 100 (1982), pp. 5-171

[5] E. Bilgin, Classes of some hypersurfaces in the Grothendieck ring of varieties, PhD thesis, Universität Duisburg–Essen, 2011.

[6] F. Brown, O. Schnetz, A K3 in $ϕ^{4}$ , Preprint, 2011, Duke Math. J., in press.

[7] Colliot-Thélène, J.-L. Birational invariants, purity and the Gersten conjecture, Santa Barbara, 1992 (Jacob, W.; Rosenberg, A., eds.) (Proc. Sympos. Pure Math.), Volume vol. 58, Part I (1995), pp. 1-64

[8] Colliot-Thélène, J.-L.; Ojanguren, M. Variétés unirationnelles non rationnelles : au-delà de lʼexemple dʼArtin et Mumford, Invent. Math., Volume 97 (1989), pp. 141-158

[9] Colliot-Thélène, J.-L.; Sansuc, J.-J. La descente sur les variétés rationnelles, Duke Math. J., Volume 54 (1987) no. 2, pp. 375-492

[10] Colliot-Thélène, J.-L.; Sansuc, J.-J. The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, pp. 113-186

[11] J.-L. Colliot-Thélène, O. Wittenberg, Groupe de Brauer et point entiers de deux familles de surfaces cubiques affines, Preprint, 2011, Amer. J. Math., in press.

[12] Grothendieck, A. Le groupe de Brauer, I, II, III, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, Mason et North-Holland, 1968

[13] Kollár, J. Conics in the Grothendieck ring, Adv. Math., Volume 198 (2005) no. 1, pp. 27-35

[14] Larsen, M.; Lunts, V.A. Motivic measures and stable birational geometry, Mosc. Math. J., Volume 3 (2003) no. 1, pp. 85-95

[15] X. Liao, Stable birational equivalence and geometric Chevalley–Warning, Preprint, 2011.

[16] Manin, Yu.I. Cubic Forms – Algebra, Geometry, Arithmetic, North-Holland, 1986

[17] Morel, F. The stable $A^{1}$ -connectivity theorems, K-theory, Volume 35 (2005), pp. 1-68

[18] Morel, F. $A^{1}$ -Algebraic Topology over a Field, Lecture Notes in Math., vol. 2052, Springer-Verlag, 2012

[19] Morel, F.; Voevodsky, V. $A^{1}$ -homotopy theory of schemes, Publ. Math. Inst. Hautes Etudes Sci., Volume 90 (2001), pp. 45-143

Cité par Sources :

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