Nori 1-motives

Abstract

Let EHM be Nori’s category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM$_{1}$$\subset$EHM generated by the i-th relative homology of pairs of varieties for i$\in${0,1}. We show that EHM${1}$ is naturally equivalent to the abelian category $^{t}$M${1}$ of 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize $^{t}$M${1}$ 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.