Abstract
Let D be a central simple algebra of prime degree over a field
and let E be an
Funding source: Agence Nationale de la Recherche
Award Identifier / Grant number: ANR-12-BL01-0005
Funding source: National Science and Engineering Board of Canada
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, ĂlĂ©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 BiaĆynickiâBirula, Invent. Math. 161 (2005), 91â111. 10.1007/s00222-004-0419-7Search in Google Scholar
[5] J.-L. Brylinski, DĂ©composition simpliciale dâun rĂ©seau, invariante par un groupe fini dâautomorphismes, C. R. Acad. Sci. Paris SĂ©r. A-B 288 (1979), no. 2, A137âA139. Search in Google Scholar
[6] V. Chernousov and A. Merkurjev, Motivic decomposition of projective homogeneous varieties and the KrullâSchmidt theorem, Transform. Groups 11 (2006), no. 3, 371â386. 10.1007/s00031-005-1114-5Search in Google Scholar
[7] J.-L. Colliot-ThĂ©lĂšne, D. Harari and A.âN. Skorobogatov, Compactification Ă©quivariante dâun tore (dâaprĂšs Brylinski et KĂŒnnemann), Expo. Math. 23 (2005), no. 2, 161â170. 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. Ăc. Norm. SupĂ©r. (4) 7 (1974), 53â88. 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 dâAzumaya et interprĂ©tations diverses, Dix exposĂ©s sur la cohomologie des schĂ©mas, North-Holland, Amsterdam (1968), 46â66. Search in Google Scholar
[15] M. Huruguen, Toric varieties and spherical embeddings over an arbitrary field, J. Algebra 342 (2011), 212â234. 10.1016/j.jalgebra.2011.05.031Search in Google Scholar
[16] N.âA. Karpenko, Characterization of minimal Pfister neighbors via Rost projectors, J. Pure Appl. Algebra 160 (2001), no. 2â3, 195â227. 10.1016/S0022-4049(00)00064-5Search in Google Scholar
[17] N.âA. Karpenko, Hyperbolicity of orthogonal involutions, Doc. Math. Extra Vol. (2010), 371â389. 10.4171/dms/5/11Search in Google Scholar
[18] N.âA. Karpenko, Upper motives of algebraic groups and incompressibility of SeveriâBrauer varieties, J. reine angew. Math. 677 (2013), 179â198. 10.1515/crelle.2012.011Search in Google Scholar
[19] N.âA. Karpenko and A.âS. Merkurjev, On standard norm varieties, Ann. Sci. Ăc. Norm. SupĂ©r. (4) 46 (2013), no. 1, 175â214. 10.24033/asens.2187Search in Google Scholar
[20] A. Merkurjev, Adams operations and the BrownâGerstenâQuillen spectral sequence, Quadratic forms, linear algebraic groups, and cohomology, Dev. Math. 18, Springer, New York (2010), 305â313. 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, 323â340. 10.1016/j.jpaa.2005.06.021Search in Google Scholar
[22] I.âA. Panin, Splitting principle and K-theory of simply connected semisimple algebraic groups, Algebra i Analiz 10 (1998), 88â131. Search in Google Scholar
[23]
D. Quillen,
Higher algebraic
[24] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), 319â393. 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 MerkurjevâSuslin variety, J. reine angew. Math. 617 (2008), 153â167. 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.âA. Suslin,
[29] A.âA. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1â2, 245â276. 10.1016/j.jpaa.2005.12.012Search in Google Scholar
[30] A. Vishik and K. Zainoulline, Motivic splitting lemma, Doc. Math. 13 (2008), 81â96. 10.4171/dm/242Search in Google Scholar
[31]
V. Voevodsky,
On motivic cohomology with
[32] V.âE. VoskresenskiÄ, 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
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Articles in the same Issue
- Frontmatter
- Equivariant basic cohomology of Riemannian foliations
- Motivic decomposition of compactifications of certain group varieties
- A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (â*)2
- Associated forms and hypersurface singularities: The binary case
- Stratified-algebraic vector bundles
- Best possible rates of distribution of dense lattice orbits in homogeneous spaces
- K-homological finiteness and hyperbolic groups
- Involutions of varieties and Rostâs degree formula
- Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
- Addendum to âSingular equivariant asymptotics and Weylâs lawâ
Articles in the same Issue
- Frontmatter
- Equivariant basic cohomology of Riemannian foliations
- Motivic decomposition of compactifications of certain group varieties
- A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (â*)2
- Associated forms and hypersurface singularities: The binary case
- Stratified-algebraic vector bundles
- Best possible rates of distribution of dense lattice orbits in homogeneous spaces
- K-homological finiteness and hyperbolic groups
- Involutions of varieties and Rostâs degree formula
- Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay
- Addendum to âSingular equivariant asymptotics and Weylâs lawâ