Nori 1-motives

Let EHM be Nori’s category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM1⊂EHM generated by the i-th relative homology of pairs of varieties for i∈{0,1}. We show that EHM1 is naturally equivalent to the abelian category tM1 of 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize tM1 as the universal abelian category obtained, using Nori’s formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori’s sense.