Home Motivic decomposition of compactifications of certain group varieties
Article
Licensed
Unlicensed Requires Authentication

Motivic decomposition of compactifications of certain group varieties

  • Nikita A. Karpenko EMAIL logo and Alexander S. Merkurjev
Published/Copyright: May 21, 2016

Abstract

Let D be a central simple algebra of prime degree over a field and let E be an 𝐒𝐋1⁡(D)-torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Award Identifier / Grant number: ANR-12-BL01-0005

Award Identifier / Grant number: Discovery Grant

Funding source: NSF

Award Identifier / Grant number: DMS #1160206

Funding statement: The first author acknowledges a partial support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005; his work has been also supported by a Discovery Grant from the National Science and Engineering Board of Canada. The work of the second author has been supported by the NSF grant DMS #1160206 and the Guggenheim Fellowship.

Acknowledgements

We thank Michel Brion for teaching us the theory of equivariant compactifications. We also thank Markus Rost and Kirill Zainoulline for helpful information.

References

[1] H. Bass, Algebraic K-theory, W. A. Benjamin, New York 1968. Search in Google Scholar

[2] N. Bourbaki, ÉlĂ©ments de mathĂ©matique. AlgĂšbre commutative. Chapitre 8. Dimension. Chapitre 9. Anneaux locaux noethĂ©riens complets, Masson, Paris 1983. Search in Google Scholar

[3] M. Brion and S. Kumar, Frobenius splitting methods in geometry and representation theory, Progr. Math. 231, BirkhÀuser, Boston 2005. 10.1007/b137486Search in Google Scholar

[4] P. Brosnan, On motivic decompositions arising from the method of BiaƂynicki–Birula, Invent. Math. 161 (2005), 91–111. 10.1007/s00222-004-0419-7Search in Google Scholar

[5] J.-L. Brylinski, DĂ©composition simpliciale d’un rĂ©seau, invariante par un groupe fini d’automorphismes, C. R. Acad. Sci. Paris SĂ©r. A-B 288 (1979), no. 2, A137–A139. Search in Google Scholar

[6] V. Chernousov and A. Merkurjev, Motivic decomposition of projective homogeneous varieties and the Krull–Schmidt theorem, Transform. Groups 11 (2006), no. 3, 371–386. 10.1007/s00031-005-1114-5Search in Google Scholar

[7] J.-L. Colliot-ThĂ©lĂšne, D. Harari and A. N. Skorobogatov, Compactification Ă©quivariante d’un tore (d’aprĂšs Brylinski et KĂŒnnemann), Expo. Math. 23 (2005), no. 2, 161–170. 10.1016/j.exmath.2005.01.016Search in Google Scholar

[8] M. Demazure, DĂ©singularisation des variĂ©tĂ©s de Schubert gĂ©nĂ©ralisĂ©es, Ann. Sci. Éc. Norm. SupĂ©r. (4) 7 (1974), 53–88. 10.24033/asens.1261Search in Google Scholar

[9] R. Elman, N. Karpenko and A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society, Providence 2008. 10.1090/coll/056Search in Google Scholar

[10] S. Evens and B. Jones, On the wonderful compactification, preprint (2008), http://arxiv.org/abs/0801.0456v1. Search in Google Scholar

[11] W. Fulton, Intersection theory, Springer, Berlin 1984. 10.1007/978-3-662-02421-8Search in Google Scholar

[12] W. Fulton, Introduction to toric varieties, Ann. of Math. Stud. 131, Princeton University Press, Princeton 1993. 10.1515/9781400882526Search in Google Scholar

[13] R. Garibaldi, A. Merkurjev and J.-P. Serre, Cohomological invariants in Galois cohomology, American Mathematical Society, Providence 2003. 10.1090/ulect/028Search in Google Scholar

[14] A. Grothendieck, Le groupe de Brauer. I. AlgĂšbres d’Azumaya et interprĂ©tations diverses, Dix exposĂ©s sur la cohomologie des schĂ©mas, North-Holland, Amsterdam (1968), 46–66. Search in Google Scholar

[15] M. Huruguen, Toric varieties and spherical embeddings over an arbitrary field, J. Algebra 342 (2011), 212–234. 10.1016/j.jalgebra.2011.05.031Search in Google Scholar

[16] N. A. Karpenko, Characterization of minimal Pfister neighbors via Rost projectors, J. Pure Appl. Algebra 160 (2001), no. 2–3, 195–227. 10.1016/S0022-4049(00)00064-5Search in Google Scholar

[17] N. A. Karpenko, Hyperbolicity of orthogonal involutions, Doc. Math. Extra Vol. (2010), 371–389. 10.4171/dms/5/11Search in Google Scholar

[18] N. A. Karpenko, Upper motives of algebraic groups and incompressibility of Severi–Brauer varieties, J. reine angew. Math. 677 (2013), 179–198. 10.1515/crelle.2012.011Search in Google Scholar

[19] N. A. Karpenko and A. S. Merkurjev, On standard norm varieties, Ann. Sci. Éc. Norm. SupĂ©r. (4) 46 (2013), no. 1, 175–214. 10.24033/asens.2187Search in Google Scholar

[20] A. Merkurjev, Adams operations and the Brown–Gersten–Quillen spectral sequence, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, Springer, New York (2010), 305–313. 10.1007/978-1-4419-6211-9_19Search in Google Scholar

[21] A. Nenashev and K. Zainoulline, Oriented cohomology and motivic decompositions of relative cellular spaces, J. Pure Appl. Algebra 205 (2006), no. 2, 323–340. 10.1016/j.jpaa.2005.06.021Search in Google Scholar

[22] I. A. Panin, Splitting principle and K-theory of simply connected semisimple algebraic groups, Algebra i Analiz 10 (1998), 88–131. Search in Google Scholar

[23] D. Quillen, Higher algebraic K-theory. I, Algebraic K-theory I, Lecture Notes in Math. 341, Springer, Berlin (1973), 85–147. 10.1007/BFb0067053Search in Google Scholar

[24] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), 319–393. 10.4171/dm/16Search in Google Scholar

[25] M. Rost, On the basic correspondence of a splitting variety, preprint (2006), www.math.uni-bielefeld.de/~rost/basic-corr.html. Search in Google Scholar

[26] N. Semenov, Motivic decomposition of a compactification of a Merkurjev–Suslin variety, J. reine angew. Math. 617 (2008), 153–167. 10.1515/CRELLE.2008.028Search in Google Scholar

[27] E. Shinder, On Motives of algebraic groups associated to division algebras, ProQuest LLC, Ann Arbor 2011; Ph.D. thesis, Northwestern University. Search in Google Scholar

[28] A. A. Suslin, K-theory and K-cohomology of certain group varieties, Algebraic K-theory, Adv. Soviet Math. 4, American Mathematical Society, Providence (1991), 53–74. 10.1090/advsov/004/05Search in Google Scholar

[29] A. A. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1–2, 245–276. 10.1016/j.jpaa.2005.12.012Search in Google Scholar

[30] A. Vishik and K. Zainoulline, Motivic splitting lemma, Doc. Math. 13 (2008), 81–96. 10.4171/dm/242Search in Google Scholar

[31] V. Voevodsky, On motivic cohomology with â„€/l-coefficients, Ann. of Math. (2) 174 (2011), no. 1, 401–438. 10.4007/annals.2011.174.1.11Search in Google Scholar

[32] V. E. Voskresenskiĭ, Algebraic groups and their birational invariants, Transl. Math. Monogr. 179, American Mathematical Society, Providence 1998. Search in Google Scholar

[33] S. Yagunov, On some differentials in the motivic cohomology spectral sequence, MPIM Preprint 153, MPIM, Bonn 2007. Search in Google Scholar

Received: 2014-02-22
Published Online: 2016-05-21
Published in Print: 2018-12-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2016-0015/html
Scroll to top button