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

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

2007-06-29

arxiv.org/abs/0706.4354v2

Mathematics

K-Theory and Homology

13 pages, The previous version was entitled `A counterexample to a conjecture of Somekawa'

Scientific paper

We construct an example of a torus $T$ over a field $K$ for which the Galois symbol $K(K; T,T)/n K(K; T,T) \to H^2(K, T[n]\otimes T[n])$ 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 $M(T\times T)$ gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

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