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

Abstract

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 ${\mathrm{Ext}}_{{\mathcal{M}}_k}^r(\mathbb{Z},G_1[-1]\otimes ...\otimes G_r[-1])$ 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 ${\mathrm{DM}}_{-}^{eff}(k)$ .

29 pages

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