Somekawas K-groups and Voevodskys Hom groups (preliminary version)

We construct a surjective homomorphism from Somekawa’s K-group associated to a finite collection of semi-abelian varieties over a perfect field to a corresponding Hom group in Voevodsky’s triangulated category of effective motivic complexes.

💡 Research Summary

The paper establishes a concrete link between Somekawa’s K‑groups, which generalize Milnor K‑theory to arbitrary finite families of semi‑abelian varieties, and the Hom‑groups that appear in Voevodsky’s triangulated category of effective motivic complexes, DM_{-}^{eff}(k). Working over a perfect base field k, the author first recalls the definition of the Somekawa K‑group K(k;G_1,…,G_r) as the quotient of the free abelian group generated by symbols {x_1,…,x_r} with x_i∈G_i(L) (L/k finite) by the relations encoding Weil reciprocity and projection formula. Parallel to this, each semi‑abelian variety G_i is regarded as a sheaf with transfers G_i^{tr}, and its Suslin complex C_∗(G_i^{tr}) is formed; the motive M(G_i) is defined as the image of this complex in DM_{-}^{eff}(k).

The central construction is a homomorphism
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