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A going down theorem for Grothendieck Chow motives
Let X be a geometrically split, geometrically irreducible variety over a
field F satisfying Rost nilpotence principle. Consider a field extension E/F
and a finite field K. We provide in this note a motivic tool giving sufficient
conditions for so-called outer motives of direct summands of the Chow motive of
X_E with coefficients in K to be lifted to the base field. This going down
result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete
classification of the motivic decompositions of projective homogeneous
varieties of inner type E_6 and to answer a conjecture of Rost and Springer.Comment: Final version of the manuscrip
Classification of upper motives of algebraic groups of inner type A_n
Let A, A' be two central simple algebras over a field F and \mathbb{F} be a
finite field of characteristic p. We prove that the upper indecomposable direct
summands of the motives of two anisotropic varieties of flags of right ideals
X(d_1,...,d_k;A) and X(d'_1,...,d'_s;A') with coefficients in \mathbb{F} are
isomorphic if and only if the p-adic valuations of gcd(d_1,...,d_k) and
gcd(d'_1,..,d'_s) are equal and the classes of the p-primary components A_p and
A'_p of A and A' generate the same group in the Brauer group of F. This result
leads to a surprising dichotomy between upper motives of absolutely simple
adjoint algebraic groups of inner type A_
Motivic decompositions of projective homogeneous varieties and change of coefficients
We prove that under some assumptions on an algebraic group ,
indecomposable direct summands of the motive of a projective -homogeneous
variety with coefficients in remain indecomposable if the ring
of coefficients is any field of characteristic . In particular for any
projective -homogeneous variety , the decomposition of the motive of
in a direct sum of indecomposable motives with coefficients in any finite field
of characteristic corresponds to the decomposition of the motive of
with coefficients in . We also construct a counterexample to this
result in the case where is arbitrary
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