Summary
Let G be a semisimple affine algebraic group over a field F. Assuming that G becomes of inner type over some finite field extension of F of degree a power of a prime p, we investigate the structure of the Chow motives with coefficients in a finite field of characteristic p of the projective G-homogeneous varieties. The complete motivic decomposition of any such variety contains one specific summand, which is the most understandable among the others and which we call the upper indecomposable summand of the variety. We show that every indecomposable motivic summand of any projective G-homogeneous variety is isomorphic to a shift of the upper summand of some (other) projective G-homogeneous variety. This result is already known (and has applications) in the case of G of inner type and is new for G of outer type (over F).
2010 Mathematics subject classification. 14L17, 14C25.
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Brosnan, P.: On motivic decompositions arising from the method of Białynicki-Birula. Invent. Math. 161(1), 91–111 (2005)
Chernousov, V., Gille, S., Merkurjev, A.: Motivic decomposition of isotropic projective homogeneous varieties. Duke Math. J. 126(1), 137–159 (2005)
Chernousov, V., Merkurjev, A.: Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem. Transform. Groups 11(3), 371–386 (2006)
Elman, R., Karpenko, N., Merkurjev, A.: The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol.56. American Mathematical Society, Providence, RI (2008)
Karpenko, N.A.: Upper motives of algebraic groups and incompressibility of Severi-Brauer varieties. Linear Algebraic Groups and Related Structures (preprint server) 333 (2009, Apr 3, revised: 2009, Apr 24), 18 pages
Tits, J.: Classification of algebraic semisimple groups. In: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pp. 33–62. American Mathematical Society, Providence, R.I. (1966)
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Karpenko, N.A. (2010). Upper Motives of Outer Algebraic Groups. In: Colliot-Thélène, JL., Garibaldi, S., Sujatha, R., Suresh, V. (eds) Quadratic Forms, Linear Algebraic Groups, and Cohomology. Developments in Mathematics, vol 18. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6211-9_15
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DOI: https://doi.org/10.1007/978-1-4419-6211-9_15
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