# Nori 1-motives

Ayoub, Joseph; Barbieri-Viale, Luca (2015). Nori 1-motives. Mathematische Annalen, 361(1-2):367-402.

## Abstract

Let EHM be Nori’s category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM$_{1}$$\subsetEHM generated by the i-th relative homology of pairs of varieties for i\in{0,1}. We show that EHM{1} is naturally equivalent to the abelian category ^{t}M{1} of 1-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize ^{t}M{1} as the universal abelian category obtained, using Nori’s formalism, from the Betti representation of an explicit diagram of curves. Secondly, we obtain a conceptual proof of a theorem of Vologodsky on realizations of 1-motives. Thirdly, we verify a conjecture of Deligne on extensions of 1-motives in the category of mixed realizations for those extensions that are effective in Nori’s sense. Let EHM be Nori’s category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory EHM_{1}$$\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.

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2015 14 Jan 2016 12:42 05 Apr 2016 19:19 Springer 0025-5831 https://doi.org/10.1007/s00208-014-1069-8
Permanent URL: https://doi.org/10.5167/uzh-111635

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