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

Abstract

We construct an example of a torus $T$ over a field $K$ for which the Galois symbol $K(K;T,T)/nK(K;T,T)\to H^2(K,T[n]\otimes T[n])$ 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).

arXiv comment13 pages, The previous version was entitled `A counterexample to a conjecture of Somekawa'

Cited by in corpus (1)

• Milnor $K$-group attached to a torus and Birch-Tate conjecture