Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-04T19:09:11.814Z Has data issue: false hasContentIssue false
  • English
  • Français

On finite dimensionality of mixed Tate motives

Published online by Cambridge University Press:  04 September 2008

Shahram Biglari
Shahram Biglari
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany, [email protected].
Get access

Abstract

We prove a few results concerning the notion of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology groups defined by means of the Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ–ring structure on K0(D).

Keywords

Tate motiveGrothendieck group
Type
Research Article
Information
Journal of K-Theory , Volume 4 , Issue 1 , August 2009 , pp. 145 - 161
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.André, Y.. Motifs de dimension finie (d'après s.-i. kimura, p. o'sullivan…). Astérisque, 299 (2005), Séminaire Bourbaki. Vol. 2003/2004, Exp. No. 929, 115–145Google Scholar
2.André, Y., Kahn, B., and O'Sullivan, P.. Nilpotence, radicaux et structures monoïdales. Rend. Sem. Mat. Univ. Padova 108 (2002), 107–291, Erratum: 113 (2005),125–128Google Scholar
3.Beĭlinson, A. A., Ginsburg, V. A., and Schechtman, V. V.. Koszul duality. J. Geom. Phys., 5(3) (1988), 317–350CrossRefGoogle Scholar
4.Biglari, S.. A Künneth formula in tensor triangulated categories. J. Pure Appl. Algebra, 210(3) (2007), 645–650CrossRefGoogle Scholar
5.Deligne, P.. Catégories tensorielles. Mosc. Math. J., 2(2) (2002), 227–248CrossRefGoogle Scholar
6.Fulton, W. and Harris, J.. Representation theory. Graduate Texts in Mathematics 129, Springer-Verlag, 1991Google Scholar
7.Guletskiĭ, V.. Finite-dimensional objects in distinguished triangles. J. Number Theory, 119(1) (2006), 99–127CrossRefGoogle Scholar
8.Huber, A. and Kahn, B.. The slice filtration and mixed Tate motives. Compos. Math. 142(4) (2006), 907–936CrossRefGoogle Scholar
9.James, G. and Kerber, A.. The representation theory of the symmetric group. Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing Co., Reading, Mass., 1981Google Scholar
10.Kapranov, M.. The elliptic curve in the s-duality theory and Eisenstein series for Kac-Moody groups. arXiv:math.AG/0001005Google Scholar
11.Kimura, S.-I.. Chow groups are finite dimensional, in some sense. Math. Ann. 331(1) (2005), 173–201CrossRefGoogle Scholar
12.Knutson, D.. λ-rings and the representation theory of the symmetric group. Lecture Notes in Mathematics 308, Springer-Verlag, 1973Google Scholar
13.Levine, M.. Tate motives and the vanishing conjectures for algebraic K-theory. in Algebraic K-theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 167–188CrossRefGoogle Scholar
14.Levine, M.. Mixed motives. Mathematical Surveys and Monographs 57, AMS, 1998Google Scholar
15.Mazza, C.. Schur functors and motives. K-Theory 33(2) (2004), 89–106CrossRefGoogle Scholar
16.Mazza, C., Voevodsky, V., and Weibel, C.. Lecture notes on motivic cohomology. Clay Mathematics Monographs 2, AMS, Providence, RI, 2006Google Scholar
17.Rivano, N. Saavedra. Catégories Tannakiennes. Lecture Notes in Mathematics 265, Springer-Verlag, 1972Google Scholar
18.Voevodsky, V.. Open problems in the motivic stable homotopy theory. I. in: Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., vol. 3, Int. Press, Somerville, MA, 2002, pp. 3–34Google Scholar
19.Voevodsky, V.. Triangulated categories of motives over a field. in: Cycles, transfers, and motivic homology theories, Ann. of Math. Stud. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238Google Scholar