Abstract
Let \(\mathsf{{ EHM}}\) be Nori’s category of effective homological mixed motives. In this paper, we consider the thick abelian subcategory \(\mathsf{{ EHM}}_1\subset \mathsf{{ EHM}}\) generated by the \(i\)-th relative homology of pairs of varieties for \(i\in \{0,1\}\). We show that \(\mathsf{{ EHM}}_1\) is naturally equivalent to the abelian category \({}^t\mathcal {M}_1\) of \(1\)-motives with torsion; this is our main theorem. Along the way, we obtain several interesting results. Firstly, we realize \({}^t\mathcal {M}_1\) 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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References
Arapura, D.: An abelian category of motivic sheaves. Adv. Math. 233(1), 135–195 (2013)
Ayoub, J.: The \(n\)-motivic \(t\)-structure for \(n=0, 1\) and \(2\). Adv. Math. 226(1), 111–138 (2011)
Ayoub, J., Barbieri-Viale, L.: 1-motivic sheaves and the Albanese functor. J. Pure Appl. Algebra 213(5), 809–839 (2009)
Barbieri-Viale, L., Kahn, B.: On the derived category of 1-motives (2014, preprint)
Barbieri-Viale, L., Rosenschon, A., Saito, M.: Deligne’s conjecture on \(1\)-motives. Ann. Math. 158(2), 593–633 (2003)
Barbieri-Viale, L., Srinivas, V.: Albanese and Picard 1-motives. Mémoires Société Mathématique de France, vol. 87, Paris (2001)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21. Springer, Berlin (1990)
Bruguières, A.: On a tannakian theorem due to Nori (2004, preprint)
Deligne, P.: Théorie de Hodge: III. Publication Mathématiques de l’I.H.É.S. 44, 5–77 (1974)
Deligne, P.: Le groupe fondamental de la droite projective moins trois points, Galois groups over \(\mathbb{Q}\) (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., vol. 16. Springer, New York, pp. 79–297 (1989)
Fakhruddin, N.: Notes of Nori’s Lectures on Mixed Motives. TIFR, Mumbai (2000)
Huber, A.: Realizations of Voevodsky’s motives. J. Algebr. Geom. 9, 755–799 (2000) [Corrigendum: J. Algebr. Geom. 13, 195–207 (2004)]
Huber, A.: Mixed motives and their realization in derived categories, Lect. Notes in Mathematics, vol. 1604. Springer, Berlin (1995)
Huber, A., Müller-Stach, S.: On the relation between Nori motives and Kontsevich periods. (2014, preprint). arXiv:1105.0865v5 [math.AG]
Levine, M.: Mixed Motives. In: Friedlander, E.M., Grayson, D.R. (eds.) Part II.5 of Handbook of K-theory. Springer, Berlin (2005)
Mazza, C., Voevodsky, V., Weibel, C.: Lecture notes on motivic cohomology. Clay Mathematics Monographs, American Mathematical Society, Providence; Clay Mathematics Institute, Cambridge, xiv+216 pp (2006)
Orgogozo, F.: Isomotifs de dimension inférieure ou égale à un. Manuscr. Math. 115(3), 339–360 (2004)
Voevodsky, V.: Triangulated categories of motives over a field, in cycles, transfers, and motivic homology theories, Annals of Math. Studies, vol. 143. Princeton Univ. Press, Princeton
Vologodsky, V.: Hodge realizations of 1-motives and the derived Albanese. J. K-Theory 10(2), 371–412 (2012)
Acknowledgments
We would like to thank M. Nori for providing some unpublished material regarding his work. We also thank D. Arapura for some helpful conversations on Nori’s work. We are grateful to the referee whose remarks lead us to include Theorem 4.3. J. Ayoub was supported in part by the Swiss National Science Foundation (NSF), Grant No. 200021-124737/1. L. Barbieri-Viale acknowledges partial support of the IHÉS who also provided hospitality and excellent working conditions.
