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Relatively unramified elements in cycle modules

Published online by Cambridge University Press:  11 April 2011

Bruno Kahn
Bruno Kahn
Affiliation:
Institut de Mathématiques de Jussieu, UMR 7586, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France, [email protected]
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Abstract

In a recent paper, Merkurjev showed that for a smooth proper variety X over a field k, the functor M* ↦ A0(X, M0) from cycle modules to abelian groups is corepresented by a cycle module constructed on the Chow group of 0-cycles of X. We show that if “proper” is relaxed, the result still holds by replacing the Chow group of 0-cycles by the 0-th Suslin homology group of X.

Keywords

Cycle modulesunramified cohomologymotivic homology
Type
Research Article
Information
Journal of K-Theory , Volume 7 , Issue 3 , June 2011 , pp. 409 - 427
Copyright
Copyright © ISOPP 2011

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References

1.Beilinson, A., Vologodsky, V.A DG guide to Voevodsky's motives, Geom. funct. anal. 17 (2007), 1709–1787.CrossRefGoogle Scholar
2.Bloch, S., Ogus, A.Gerten's conjecture and the homology of schemes, Ann. Sci. ÉNS 7 (1974), 181–201 (1975).Google Scholar
3.Bloch, S.The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), 537–568.Google Scholar
4.Cisinski, D.-C., Déglise, F.Local and stable homological algebra in Grothendieck abelian categories, Homology, Homotopy Appl. 11 (2009), 219–260.CrossRefGoogle Scholar
5.Déglise, F.Transferts sur les groupes de Chow à coefficients, Math. Zeit. 252 (2006), 315–343.CrossRefGoogle Scholar
6.Déglise, F.Motifs génériques, Rend. Mat. Sem. Univ. Padova 119 (2008), 173–244.CrossRefGoogle Scholar
7.Déglise, F.Modules homotopiques, preprint, 2009.Google Scholar
8.Esnault, H., Kahn, B., Levine, M., Viehweg, E.The Arason invariant and mod 2 algebraic cycles, J. Amer. Math. Soc. 11 (1988), 73–118.CrossRefGoogle Scholar
9.Hartshorne, R. Algebraic geometry, Springer, 1977.CrossRefGoogle Scholar
10.Merkurjev, A. S.Unramified elements in cycle modules, J. London Math. Soc. 78 (2008), 51–64.CrossRefGoogle Scholar
11.Morel, F.On the motivic π0 of the sphere spectrum, in Axiomatic, enriched and motivic homotopy theory (Greenlees, J.P.C, ed.), NATO Sci. Series II. (Math. Phys. Chem.) 131, Kluwer, 2004, 219–260.Google Scholar
12.Rost, M.Chow groups with coefficients, Doc. Math. 1 (1996), 319–393.CrossRefGoogle Scholar
13.Friedlander, E., Voevodsky, V.Bivariant cycle cohomology, in Cycles, transfers and motivic cohomology theories, Ann. of Math. Studies 143, Princeton, 2000, 138–187.Google Scholar
14.Suslin, A., Voevodsky, V.Bloch-Kato conjecture and motivic cohomology with finite coefficients, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 117–189, NATO Sci. Ser. C Math. Phys. Sci. 548, Kluwer, 2000.Google Scholar
15.Voevodsky, V.Triangulated categories of motives over a field, in Cycles, transfers and motivic cohomology theories, Ann. of Math. Studies 143, Princeton, 2000, 188–238.Google Scholar
16.Voevodsky, V.Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Int. Math. Res. Not. 2002, 351–355.CrossRefGoogle Scholar
17.Voevodsky, V.Cancellation theorem, Doc. Math. Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010) 671–685.Google Scholar