Mathematische Annalen

, Volume 331, Issue 1, pp 173–201

Chow groups are finite dimensional, in some sense

Article

Abstract.

When S is a surface with p g (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 p g (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 p g (S)=0) under the assumption that the Chow motive of S is finite dimensional.

Keywords

Strong Property Chow Group Finite Dimensionality Chow Motive
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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)
3. 3.
Bloch, S.: Some elementary theorems about algebraic cycles on abelian varieties. Invent. Math. 37, 215–228 (1976)
4. 4.
Bloch, S., Kas, A., Lieberman, D.: Zero cycles on surfaces with pg=0. Compositio Math. 33, 135–145 (1976)
5. 5.
Deninger, C., Murre, J.: Motivic decomposition of Abelian schemes and the Fourier transform. J. Reine. Angew. Math. 422, 201–219 (1991)
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)
10. 10.
11. 11.
Kimura, S.: A cohomological characterization of Alexander schemes. Invent. math. 137, 575–611 (1999)
12. 12.
Kimura, S., Vistoli, A.: Chow rings of infinite symmetric products. Duke Math. J. 85, 411–430 (1996)
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)
17. 17.
Roitman, A.A.: The torsion of the group of 0-cycles modulo rational equivalence. Ann. Math. 111, 553–569 (1980)
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)