A counterexample to generalizations of the Milnor-Bloch-Kato conjecture , by Michael Spiess and Takao Yamazaki

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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