Chow groups of products of Severi-Brauer varieties and invariants of degree $3$
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Abstract:
We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension $2$ Chow groups of a product of Severi-Brauer varieties. In particular, for any $n\geq 2$ we completely determine the degree $3$ invariants of a split semisimple group, the quotient of $(\mathbf {SL}_{2})^{n}$ by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.References
- S. Baek, Codimension 2 cycles on products of projective homogeneous surfaces, Preprint available at http://arxiv.org/abs/1307.0669.
- Sanghoon Baek and Alexander Merkurjev, Invariants of simple algebras, Manuscripta Math. 129 (2009), no. 4, 409–421. MR 2520892, DOI 10.1007/s00229-009-0265-4
- Hernando Bermudez and Anthony Ruozzi, Degree 3 cohomological invariants of split simple groups that are neither simply connected nor adjoint, J. Ramanujan Math. Soc. 29 (2014), no. 4, 465–481. MR 3284049
- Sam Blinstein and Alexander Merkurjev, Cohomological invariants of algebraic tori, Algebra Number Theory 7 (2013), no. 7, 1643–1684. MR 3117503, DOI 10.2140/ant.2013.7.1643
- Skip Garibaldi, Cohomological invariants: exceptional groups and spin groups, Mem. Amer. Math. Soc. 200 (2009), no. 937, xii+81. With an appendix by Detlev W. Hoffmann. MR 2528487, DOI 10.1090/memo/0937
- Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003. MR 1999383, DOI 10.1090/ulect/028
- A. Merkurjev, Degree three cohomological invariants of semisimple groups, To appear in JEMS.
- A. Merkurjev, Unramified degree three invariants of reductive groups, Preprint available at http://www.mathematik.uni-bielefeld.de/LAG/man/543.html.
- Alexander Merkurjev, Alexander Neshitov, and Kirill Zainoulline, Invariants of degree 3 and torsion in the Chow group of a versal flag, Compos. Math. 151 (2015), no. 8, 1416–1432. MR 3383162, DOI 10.1112/S0010437X14008057
- Oleg T. Izhboldin and Nikita A. Karpenko, Generic splitting fields of central simple algebras: Galois cohomology and nonexcellence, Algebr. Represent. Theory 2 (1999), no. 1, 19–59. MR 1688470, DOI 10.1023/A:1009910324736
- N. A. Karpenko, On topological filtration for Severi-Brauer varieties, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 275–277. MR 1327303, DOI 10.1007/bf02567809
- Nikita A. Karpenko, Codimension $2$ cycles on Severi-Brauer varieties, $K$-Theory 13 (1998), no. 4, 305–330. MR 1615533, DOI 10.1023/A:1007705720373
- M.-A. Knus, T. Y. Lam, D. B. Shapiro, and J.-P. Tignol, Discriminants of involutions on biquaternion algebras, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 279–303. MR 1327304
- A. S. Sivatski, The chain lemma for biquaternion algebras, J. Algebra 350 (2012), 170–173. MR 2859881, DOI 10.1016/j.jalgebra.2011.09.037
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
Additional Information
- Sanghoon Baek
- Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Republic of Korea
- MR Author ID: 875898
- Email: sanghoonbaek@kaist.ac.kr
- Received by editor(s): March 4, 2015
- Published electronically: May 3, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 1757-1771
- MSC (2010): Primary 11E72, 14F43, 14M17, 20G15
- DOI: https://doi.org/10.1090/tran/6772
- MathSciNet review: 3581218