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A counterexample to generalizations of the Milnor-Bloch-Kato conjecture

Published online by Cambridge University Press:  25 August 2009

Michael Spiess  and
Takao Yamazaki
Michael Spiess
Affiliation:
Fakultät für Mathematik, Postfach 100131, D-33501 Bielefeld, Germany, mspiess@math.uni-bielefeld.de.
Takao Yamazaki
Affiliation:
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan, ytakao@math.tohoku.ac.jp.
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Abstract

We construct an example of a torus T over a field K for which the Galois symbol K(K;T,T)/nK(K;T,T) → H2(K,T[n] ⊗ T[n]) is not injective for some n. Here K(K;T,T) is the Milnor K-group attached to T introduced by Somekawa. We show also that the motive M(T × T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

Keywords

Motivic cohomologyhigher algebraic K-theoryMilnor K-group
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 1 , August 2009 , pp. 77 - 90
Copyright
Copyright © ISOPP 2009

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References

Be.Beilinson, A.A., Talk at Fields Institute at Toronto on 21 03 2007.Google Scholar
HK.Huber, A., Kahn, B., The slice filtration and mixed Tate motives. Compos. Math. 142 (2006), 907–936.CrossRefGoogle Scholar
Ka.Kato, K., A generalization of local class field theory by using K-groups I, J. Fac. Sci. Univ. Tokyo 26 (1979), 303–376.Google Scholar
MVW.Mazza, C., Voevodsky, V. and Weibel, C., Lecture notes on motivic cohomology. Clay Mathematics Monographs 2. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006.Google Scholar
MS.Merkurjev, A.S., Suslin, A.A., K-cohomology of Severi-Brauer varieties and the norm residue homomorphism. Math. USSR Izvestiya 21 (1983), 307–340.CrossRefGoogle Scholar
Se.Serre, J.-P., Algebraic groups and class fields. Graduate Texts in Mathematics 117. Springer Verlag 1988.CrossRefGoogle Scholar
So.Somekawa, M., On Milnor K-groups attached to semi-abelian varieties. K-theory 4 (1990), 105–119.CrossRefGoogle Scholar
SV.Suslin, A.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 Acad. Publ., Dordrecht, 2000.Google Scholar
Vo1.Voevodsky, V., Triangulated categories of motives over a field. In: Voevodsky, V., Suslin, A., Friedlander, E.M.: Cycles, Transfers, and Motivic Homology Theories. Annals of Math. Studies 143, Princeton University Press, 2000, 188–238.Google Scholar
Vo2.Voevodsky, V., Motivic cohomology with ℤ/2-coefficients. Publ. Math., Inst. Hautes Étud. Sci. 98, 59–104 (2003).CrossRefGoogle Scholar