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

0706.4354Jun 29, 200713 pages, The previous version was entitled `A counterexample to a conjecture of Somekawa'

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

Michael Spiess, Takao Yamazaki

Abstract

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

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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) → H²(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.

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Claims

The paper addresses: A counterexample to generalizations of the Milnor-Bloch-Kato conjecture.
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) → H²(K, T[n]⊗ T[n]) is not injective for some n.

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