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

Michael Spiess; Takao Yamazaki

doi:10.1017/is008008014jkt066

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

Michael Spiess, Takao Yamazaki

SCI - Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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) 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. We show also that the motive M(T×T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

Original language	English
Pages (from-to)	77-90
Number of pages	14
Journal	Journal of K-Theory
Volume	4
Issue number	1
DOIs	https://doi.org/10.1017/is008008014jkt066
Publication status	Published - 2009 Aug 1

Keywords

Higher algebraic K-theory
Milnor K-group
Motivic cohomology

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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N2 - 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) 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. We show also that the motive M(T×T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

AB - 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) 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. We show also that the motive M(T×T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

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KW - Milnor K-group

KW - Motivic cohomology

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