Hostname: page-component-7dd5485656-c8zdz Total loading time: 0 Render date: 2025-10-29T18:31:36.356Z Has data issue: false hasContentIssue false
  • English
  • Français

The Brauer group and indecomposable $(2,1)$-cycles

Part of: $K$-theory in geometry (Co)homology theory

Published online by Cambridge University Press:  17 December 2015

Bruno Kahn
Bruno Kahn*
Affiliation:
IMJ-PRG, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France email bruno.kahn@imj-prg.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the torsion in the group of indecomposable $(2,1)$-cycles on a smooth projective variety over an algebraically closed field is isomorphic to a twist of its Brauer group, away from the characteristic. In particular, this group is infinite as soon as $b_{2}-{\it\rho}>0$. We derive a new insight into Roǐtman’s theorem on torsion $0$-cycles over a surface.

Keywords

motivic cohomologyBrauer groupRoǐtman’s theorem

MSC classification

Primary: 19E15: Algebraic cycles and motivic cohomology 14F22: Brauer groups of schemes

Information

Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 1041 - 1051
Copyright
© The Author 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Barbieri-Viale, L. and Kahn, B., On the derived category of 1-motives, Astérisque, to appear, arXiv:1009.1900.Google Scholar
Bloch, S., Torsion algebraic cycles and a theorem of Roitman, Compositio Math. 39 (1979), 107–127.Google Scholar
Chen, X., Doran, C., Kerr, M. and Lewis, J., Normal functions, Picard–Fuchs equations, and elliptic fibrations on K3 surfaces, J. Reine Angew. Math., doi:10.1515/crelle-2014-0085, November (2014).CrossRefGoogle Scholar
Colliot-Thélène, J.-L. and Raskind, W., K2 -cohomology and the second Chow group, Math. Ann. 270 (1985), 165–199.CrossRefGoogle Scholar
Gabber, O., Sur la torsion dans la cohomologie l-adique d’une variété, C. R. Acad. Sci. Paris 297 (1983), 179–182.Google Scholar
Gros, M. and Suwa, N., Application d’Abel-Jacobi p-adique et cycles algébriques, Duke Math. J. 57 (1988), 579–613.CrossRefGoogle Scholar
Illusie, L., Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. 12 (1979), 501–661.Google Scholar
Illusie, L. and Raynaud, M., Les suites spectrales associées au complexe de Rham-Witt, Publ. Math. Inst. Hautes Études Sci. 57 (1983), 73–212.Google Scholar
Kahn, B., Murre, J. P. and Pedrini, C., On the transcendental part of the motive of a surface, in Algebraic Cycles and Motives, London Mathematical Society Series, vol. 344 (Cambridge University Press, Cambridge, 2007), 143–202.Google Scholar
Kahn, B., Classes de cycle motiviques étales, Algebra Number Theory 6–7 (2012), 1369–1407.Google Scholar
Milne, J. S., Values of zeta functions of varieties over finite fields, Amer. J. Math. 108 (1988), 297–360.Google Scholar
Murre, J., On the motive of an algebraic surface, J. Reine Angew. Math. 409 (1990), 190–204.Google Scholar
Panin, I., Fields whose K 2 is 0. Torsion in H 1(X, K2 and CH 2(X), Zap. LOMI 116 (1982), 1011–1046.Google Scholar
Scholl, A. J., Classical motives, Proc. Sympos. Pure Math. 55(I) (1994), 163–187.Google Scholar
Jouanolou, J.-P., Systèmes projectifs l-adiques, in Séminaire de géométrie algébrique du Bois-Marie - 1965–1966 - Cohomologie l-adique et fonctions L (SGA5), Lecture Notes in Mathematics, vol. 589, Exp. V, eds Grothendieck, A. et al. (Springer, Berlin, 1977), 204–250.Google Scholar
Kleiman, S., Les théorèmes de finitude pour le foncteur de Picard, in Séminaire de géométrie algébrique du Bois-Marie - 1966–1967 - Théorie des intersections et théorème de Riemann-Roch (SGA6), Lecture Notes in Mathematics, vol. 225, Exp. XIII, eds Berthelot, P., Grothendieck, A. and Illusie, L. (Springer, Berlin, 1971), 616–666.Google Scholar
Voisin, C., Variations of Hodge structure and algebraic cycles, in Proceedings of the International Congress of Mathematicians, August 3–11, 1994, Zürich, Switzerland (Birkhäuser, 1975).Google Scholar