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

摘要

We construct an example of a torus T over a field K for which the Galois symbol K(K;T,T)K(K;T,T) H~2(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).