We construct an example of a torus $T$ over a field $K$ for which the Galoissymbol $K(K; T,T)/n K(K; T,T) o H^2(K, T[n]otimes T[n])$ is not injectivefor some $n$. Here $K(K; T,T)$ is the Milnor $K$-group attached to $T$introduced by Somekawa. We show also that the motive $M(Times T)$ gives acounterexample to another generalization of the Milnor-Bloch-Kato conjecture(proposed by Beilinson).
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机译:我们构造了一个在字段$ K $上的圆环$ T $的例子,为此,Galoissymbol $ K(K; T,T)/ n K(K; T,T) to H ^ 2(K,T [n ] otimes T [n])$对于某些$ n $不是内射的。在这里,$ K(K; T,T)$是Somekawa引入的附加到$ T $的Milnor $ K $-组。我们还表明,动机$ M(T times T)$给出了Milnor-Bloch-Kato猜想的另一种推广的反例(由Beilinson提出)。
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