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On the torsion of Chow groups of Severi-Brauer varieties

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Abstract

In this paper, we generalize a result of Karpenko on the torsion in the second quotient of the gamma filtration for Severi-Brauer varieties to higher degrees. As an application, we provide a nontrivial torsion in the Chow groups of higher codimension and the topological filtration of the associated generic variety and obtain new upper bounds for the annihilators of the torsion subgroups in the Chow groups of a large class of Severi-Brauer varieties. In particular, using the torsion in higher degrees, we show indecomposability of certain algebras.

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References

  1. S. A. Amitsur, L. H. Rowen and J.-P. Tignol, Division algebras of degree 4 and 8 with involution, Israel Journal of Mathematics 33 (1979), 133–148.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Baek, E. Neher and K. Zainoulline, Basic polynomial invariants, fundamental representations and the Chern class map, Documenta Mathematica 17 (2012), 135–150.

    MATH  MathSciNet  Google Scholar 

  3. S. Baek, K. Zainoulline and C. Zhong, On the torsion of Chow groups of twisted Spinflags, Mathematical Research Letters 20 (2013), 601–614.

    Article  MATH  MathSciNet  Google Scholar 

  4. W. Fulton, Intersection Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 2, Springer-Verlag, Berlin, 1984.

    Book  MATH  Google Scholar 

  5. W. Fulton and S. Lang, Riemann-Roch Algebra, Grundlehren der Mathematischen Wissenschaften, Vol. 277, Springer-Verlag, New York, 1985.

    MATH  Google Scholar 

  6. S. Garibaldi, A. Merkurjev and J.-P. Serre, Cohomological Invariants in Galois Cohomology, University Lecture Series, Vol. 23, American Mathematical Society, Providence, RI, 2003.

    MATH  Google Scholar 

  7. S. Garibaldi and K. Zainoulline, The gamma-filtration and the Rost invariant, Journal für die Reine und Angewandte Mathematik, to appear.

  8. N. Karpenko, On topological filtration for Severi-B rauer varieties, in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Snata Barbara, CA. 1992), Proceedings of Symposia in Pure Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 1995, pp. 275–277.

    Google Scholar 

  9. N. Karpenko, On topological filtration for Severi-Brauer varieties. II, in Mathematics in St. Petersburg, American Mathematical Society Translations, Vol. 174, American Mathematical Society, Providence, RI, 1996, pp. 45–48.

    Google Scholar 

  10. N. Karpenko, On topological filtration for triquaternion algebra, Nova Journal of Mathematics, Game Theory and Algebra 4 (1996), 161–167.

    MATH  MathSciNet  Google Scholar 

  11. N. A. Karpenko, Order of torsion in CH 4 of quadrics, Documenta Mathematica 1 (1996), 57–65.

    MATH  MathSciNet  Google Scholar 

  12. N. Karpenko, Codimension 2 cycles on Severi-Brauer varieties, K-Theory 13 (1998), 305–330.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Karpenko and A. Merkurjev, Chow groups of projective quadrics, Leningrad Mathematical Journal 2 (1991), 655–671.

    MathSciNet  Google Scholar 

  14. Yu. I. Manin, Lectures on the K-functor in algebraic geometry, Uspehi Matematičeskih Nauk 24 (1969), no. 5, 1–89.

    MATH  MathSciNet  Google Scholar 

  15. A. S. Merkurjev, Certain K-cohomology groups of Severi-Brauer varieties, in K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras (Snata Barbara, CA. 1992), Proceedings of Symposia in Pure Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 1995, pp. 319–331.

    Google Scholar 

  16. E. Peyre, Galois cohomology in degree three and homogeneous varieties, K-Theory 15 (1995), 99–145.

    Article  MathSciNet  Google Scholar 

  17. D. Quillen, Higher algebraic K-theory. I, in algebraic K-Theory, I: Higher K-Theories (Proc. Conf. Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics, Vol. 341, Springer, Berlin, 1973, pp. 85–147.

    Google Scholar 

  18. A. Schofield and M. Van den Bergh, The index of a Brauer class on a Brauer-Severi variety, Transactions of the American Mathematical Society 333 (1992), 729–739.

    Article  MATH  MathSciNet  Google Scholar 

  19. J.-P. Tignol, Algèbres indécomposables d’exposant premier, Advances in Mathematics 65 (1987), 205–228.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Vishik, On torsion elements in the Chow groups of quadrics, preprint (2000), available at https://www.maths.nottingham.ac.uk/personal/av/papers.html.

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  1. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea

    Sanghoon Baek

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  1. Sanghoon Baek
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Baek, S. On the torsion of Chow groups of Severi-Brauer varieties. Isr. J. Math. 207, 899–923 (2015). https://doi.org/10.1007/s11856-015-1166-8

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  • DOI: https://doi.org/10.1007/s11856-015-1166-8

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