Comptes Rendus
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]
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
@article{CRMATH_2012__350_11-12_613_0,
     author = {Le Dang Thi Nguyen},
     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},
}
TY  - JOUR
AU  - Le Dang Thi Nguyen
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
DO  - 10.1016/j.crma.2012.05.008
LA  - en
ID  - CRMATH_2012__350_11-12_613_0
ER  - 
%0 Journal Article
%A Le Dang Thi Nguyen
%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
%R 10.1016/j.crma.2012.05.008
%G en
%F CRMATH_2012__350_11-12_613_0
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/

[1] A. Asok, Birational invariants and A1-connectedness, Preprint, 2011, Crelleʼs J., , in press. | DOI

[2] A. Asok; F. Morel Smooth varieties up to A1-homotopy and algebraic h-cobordisms, Adv. Math., Volume 227 (2011) no. 5, pp. 1990-2058

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

[4] A.A. Beilinson; J. Bernstein; P. Deligne 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 ϕ4, Preprint, 2011, Duke Math. J., in press.

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

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

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

[10] J.-L. Colliot-Thélène; J.-J. Sansuc 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] A. Grothendieck 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] J. Kollár Conics in the Grothendieck ring, Adv. Math., Volume 198 (2005) no. 1, pp. 27-35

[14] M. Larsen; V.A. Lunts 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] Yu.I. Manin Cubic Forms – Algebra, Geometry, Arithmetic, North-Holland, 1986

[17] F. Morel The stable A1-connectivity theorems, K-theory, Volume 35 (2005), pp. 1-68

[18] F. Morel A1-Algebraic Topology over a Field, Lecture Notes in Math., vol. 2052, Springer-Verlag, 2012

[19] F. Morel; V. Voevodsky A1-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”.

Commentaires - Politique