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Tensor Structure on Smooth Motives

Published online by Cambridge University Press:  16 May 2011

Anandam Banerjee
Anandam Banerjee
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, [email protected]
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Abstract

Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

Keywords

MotivescorrespondencesDG categorypseudo-tensor structure
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 1 , February 2012 , pp. 57 - 101
Copyright
Copyright © ISOPP 2011

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References

1.Banerjee, A.: Tensor structure on Smooth Motives (2010). Preprint, available at http://arxiv.org/abs/1004.1491Google Scholar
2.Beilinson, A., Drinfeld, V.: Chiral Algebras. AMS (2004)CrossRefGoogle Scholar
3.Beilinson, A., Vologodsky, V.: A DG Guide to Voevodsky's Motives. Geometric And Functional Analysis 14(6)(2008), 1709–1787CrossRefGoogle Scholar
4.Bondarko, M. V.: Differential graded motives: weight complex, weight filtrations and spectral sequences for realizations; Voevodsky versus Hanamura. J. Inst. Math. Jussieu 8(1) (2009), 39–97CrossRefGoogle Scholar
5.Cisinski, D. C., Déglise, F.: Triangulated categories of motives (2007). Preprint, available at http://www.math.univ-paris13.fr/~deglise/docs/2009/DM.pdfGoogle Scholar
6.Friedlander, E., Voevodsky, V.: Bivariant cycle cohomology. Annals of Math. Studies 143 (2000), 138–187Google Scholar
7.Geisser, T., Levine, M.: The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky. Jour. f.d. Reine u. Ang. Math. 530 (2001), 55–103Google Scholar
8.Hanamura, M.: Mixed motives and algebraic cycles, II. Invent. Math. 158(1) (2004), 105–179CrossRefGoogle Scholar
9.Keller, B.: Deriving DG Categories. Ann. Sci. Éc. Norm. Sup. 27 (1994), 63–102. 4e sérieCrossRefGoogle Scholar
10.Levine, M.: Bloch's higher Chow groups revisited. Astérisque 226 (1994), 235–320Google Scholar
11.Levine, M.: Smooth Motives. Fields Institute Communication 56 (2009), 175–232Google Scholar
12.Toën, B.: The homotopy theory of dg-categories and derived Morita theory. Invent. Math. 167 (2007), 615–667CrossRefGoogle Scholar
13.Voevodsky, V.: Motives over Simplicial Schemes (2008). PreprintGoogle Scholar
14.Weibel, C. A.: An introduction to homological algebra. Cambridge University Press (1994)CrossRefGoogle Scholar