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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}$$\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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > General Mathematics
Language:English
Date:2015
Deposited On:14 Jan 2016 12:42
Last Modified:13 Jul 2024 01:39
Publisher:Springer
ISSN:0025-5831
OA Status:Green
Publisher DOI:https://doi.org/10.1007/s00208-014-1069-8
  • Content: Accepted Version
  • Language: English