Motivic interpretation of Milnor $K$-groups attached to Jacobian varieties

Details Motivic interpretation of Milnor $K$-groups attached to Jacobian varieties Motivic interpretation of Milnor $K$-groups attached to Jacobian varieties

2006-03-10

arxiv.org/abs/math/0603241v1

Mathematics

K-Theory and Homology

29 pages

Scientific paper

In the paper M. Somekawa, {\it{On Milnor $K$-groups attached at semi-Abelian varieties}}, K-theory, \textbf{4} (1990) p.105, Somekawa conjectures that his Milnor K-group $K(k,G_1,...,G_r)$ attached to semi-abelian varieties $G_1$,...,$G_r$ over a field $k$ is isomorphic to ${\rm Ext}_{\mathcal{M}_k}^r(\mathbb{Z},G_1[-1] \otimes ... \otimes G_r[-1])$ where $\mathcal{M}_k$ is a certain category of motives over $k$. The purpose of this note is to give remarks on this conjecture, when we take $\mathcal{M}_k$ as Voevodsky's category of motives ${\rm DM}^{eff}_{-}(k)$ .

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