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Mixed motives over k[t]/(tm+1)

Part of: $K$-theory in geometry Cycles and subschemes

Published online by Cambridge University Press:  03 January 2012

Amalendu Krishna  and
Jinhyun Park
Amalendu Krishna
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India (amal@math.tifr.res.in)
Jinhyun Park
Affiliation:
Department of Mathematical Sciences, KAIST, Yuseong-gu, Daejeon, 305-701, Republic of Korea (jinhyun@mathsci.kaist.ac.kr; jinhyun@kaist.edu)
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Abstract

For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

Keywords

Chow groupalgebraic cyclemoving lemmamotive

MSC classification

Secondary: 14C25: Algebraic cycles 19E15: Algebraic cycles and motivic cohomology

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 3 , July 2012 , pp. 611 - 657
Copyright
Copyright © Cambridge University Press 2011

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References

1.Balmer, P. and Schlichting, M., Idempotent completion of triangulated categories, J. Alg. 236(2) (2001), 819–834.Google Scholar
2.Banerjee, A., Tensor structure on smooth motives, J. K-Theory, in press (DOI:10.1017/is011003009jkt144).Google Scholar
3.Beilinson, A. and Drinfeld, V., Chiral algebras, American Mathematical Society Colloquium Publications, Volume 51 (American Mathematical Society, Providence, RI, 2004).Google Scholar
4.Bloch, S., On the tangent space to Quillen K-theory, in Algebraic K-theory, Volume I: Higher K-theories (Proceedings of a Conference at the Battelle Memorial Institute, Seattle, WA, 1972), pp. 205–210, Lecture Notes in Mathematics, Volume 341 (Springer, 1973).Google Scholar
5.Bloch, S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267–304.Google Scholar
6.Bloch, S., Algebraic cycles and the Lie algebra of mixed Tate motives, J. Am. Math. Soc. 4(4) (1991), 771–791.Google Scholar
7.Bloch, S., The moving lemma for higher Chow groups, J. Alg. Geom. 3(3) (1994), 537–568.Google Scholar
8.Bloch, S. and Esnault, H., The additive dilogarithm, Doc. Math. Extra Volume (Kazuya Kato's Fiftieth Birthday) (2003), 131–155.Google Scholar
9.Bondal, A. I. and Kapranov, M. M., Framed triangulated categories, Mat. Sb. 181(5) (1990), 669–683 (in Russian; English translation: ‘Enhanced triangulated categories’, Math. USSR Sb. 70(1) (1991), 93–107).Google Scholar
10.Fulton, W., Intersection theory, 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Folge A, Series of Modern Surveys in Mathematics, Volume 2 (Springer, 1998).Google Scholar
11.Goncharov, A., Euclidean scissor congruence groups and mixed Tate motives over dual numbers, Math. Res. Lett. 11(5–6) (2004), 771–784.CrossRefGoogle Scholar
12.Guillén, F. and Aznar, V. Navarro, Un critère d'extension des foncteurs définis sur les schémas lisses, Publ. Math. IHES 95 (2002), 1–91.CrossRefGoogle Scholar
13.Hanamura, M., Homological and cohomological motives of algebraic varieties, Invent. Math. 142 (2000), 319–149.Google Scholar
14.Hanamura, M., Mixed motives and algebraic cycles, II, Invent. Math. 158 (2004), 105–179.Google Scholar
15.Hesselholt, L., K-theory of truncated polynomial algebras, in Handbook of K-theory, Volume 1, pp. 71–110 (Springer, 2005).CrossRefGoogle Scholar
16.Krishna, A. and Levine, M., Additive higher Chow groups of schemes, J. Reine Angew. Math. 619 (2008), 75–140.Google Scholar
17.Krishna, A. and Park, J., Moving lemma for additive higher Chow groups, Alg. Num. Theory, in press.Google Scholar
18.Krishna, A. and Park, J., Additive higher Chow group of 1-cycles on fields, in preparation.Google Scholar
19.Levine, M., Mixed motives, Mathematical Surveys and Monographs, Volume 57 (American Mathematical Society, Providence, RI, 1998).Google Scholar
20.Levine, M., Smooth motives, in Motives and algebraic cycles, pp. 175–231, Fields Institute Communications, Volume 56 (American Mathematical Society, Providence, RI, 2009).Google Scholar
21.Park, J., Algebraic cycles and additive dilogarithm, Int. Math. Res. Not. 2007(18) (2007), rnm067.Google Scholar
22.Park, J., Regulators on additive higher Chow groups, Am. J. Math. 131(1) (2009), 257–276.Google Scholar
23.Rülling, K., The generalized de Rham–Witt complex over a field is a complex of zerocycles, J. Alg. Geom. 16(1) (2007), 109–169.Google Scholar
24.Soulé, C., Opérations en K-théorie algébrique, Can. J. Math. 37(3) (1985), 488–550.Google Scholar
25.Voevodsky, V., Triangulated categories of motives over a field, in Cycles, transfers, and motivic cohomology theories (ed. Voevodsky, V., Suslin, A. and Friedlander, E. M.) (Princeton University Press, 2000).Google Scholar