Mathematische Annalen

, Volume 331, Issue 1, pp 173–201 | Cite as

Chow groups are finite dimensional, in some sense

Article

Abstract.

When S is a surface with pg(S)>0, Mumford proved that its Chow group A*S is not “finite dimensional” in some sense. In this paper, we propose another definition of “finite dimensionality” for the Chow groups. Using this new definition, at least the Chow group of some surface S with pg(S)>0 (for example, the product of two curves) becomes finite dimensional. The finite dimensionality of the Chow groups follows from the finite dimensionality of the Chow motives. It turns out that the finite dimensionality of the Chow motives is a very strong property. For example, we can prove Bloch’s conjecture (representability of the Chow groups of surfaces with pg(S)=0) under the assumption that the Chow motive of S is finite dimensional.

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References

  1. 1.
    André, Y.: Motifs de dimension finie. Séminaire Bourbaki, 2004Google Scholar
  2. 2.
    Beauville, A.: Sur l’anneau de Chow d’une variété abélienne. Math. Ann. 273, 647–651 (1986)MathSciNetMATHGoogle Scholar
  3. 3.
    Bloch, S.: Some elementary theorems about algebraic cycles on abelian varieties. Invent. Math. 37, 215–228 (1976)MATHGoogle Scholar
  4. 4.
    Bloch, S., Kas, A., Lieberman, D.: Zero cycles on surfaces with pg=0. Compositio Math. 33, 135–145 (1976)MATHGoogle Scholar
  5. 5.
    Deninger, C., Murre, J.: Motivic decomposition of Abelian schemes and the Fourier transform. J. Reine. Angew. Math. 422, 201–219 (1991)MathSciNetMATHGoogle Scholar
  6. 6.
    Fulton, W.: Intersection Theory. Springer, Berlin, 1984, pp. xi+470Google Scholar
  7. 7.
    Fulton, W., Harris, J.: Representation Theory. Springer GTM 129, New York, 1991, pp. xvi+551Google Scholar
  8. 8.
    Guletskii, V.: A remark on nilpotent correspondences, K-theory preprint archive 651 (2003), http://www.math.uiuc.edu/K-theory
  9. 9.
    Kimura, S.: Fractional Intersection and Bivariant Theory. Commun. Algebra 20, 285–302 (1992)MathSciNetMATHGoogle Scholar
  10. 10.
    Kimura, S.: On Varieties whose Chow Groups have Intersection Products with Open image in new window-coefficients. Thesis, University of Chicago, 1990Google Scholar
  11. 11.
    Kimura, S.: A cohomological characterization of Alexander schemes. Invent. math. 137, 575–611 (1999)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Kimura, S., Vistoli, A.: Chow rings of infinite symmetric products. Duke Math. J. 85, 411–430 (1996)MathSciNetMATHGoogle Scholar
  13. 13.
    Kleiman, S.: The standard conjectures. In: Motives (Seattle, WA, 1991). Proc. Sympos. Pure Math. Vol 55, Amer. Math. Soc. pp. 3–20, 1994Google Scholar
  14. 14.
    Knutson, D.: λ-Rings and the Representation Theory of the Symmetric Group. Lecture Notes in Math. 308, (Springer, Berlin 1973) iv+203 ppGoogle Scholar
  15. 15.
    Mazza, C.: Schur functors and motives. K-theory preprint archive 641 (2003), http://www.math.uiuc.edu/K-theory/
  16. 16.
    Mumford, D.: Rational equivalences of 0-cycles on surfaces. J. Math. Kyoto Univ. 9, 195–204 (1969)MATHGoogle Scholar
  17. 17.
    Roitman, A.A.: The torsion of the group of 0-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)MathSciNetGoogle Scholar
  18. 18.
    Scwarzenberger, R.L.E.: Jacobians and Symmetric Products. Illinois J. Math. 7, 257–268 (1963)Google Scholar
  19. 19.
    Shermenev, A.M.: The motive of an abelian variety. Funct. Anal. 8, 47–53 (1974)Google Scholar
  20. 20.
    Voevodsky, V.: Nilpotence theorem for cycles algebraically equivalent to zero. Internat. Math. Res. Notices pp. 187–198 (1995)Google Scholar
  21. 21.
    Vistoli, A.: Intersection theory on algebraic stacks and on their moduli spaces. Invent. math. 97, 613–670 (1989)MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsHiroshima UniversityHigashi-HiroshimaJapan

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