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Have a look on the paper

F. Morel, Voevodsky's proof of Milnor's conjecture, Bull. Amer. Math. Soc. 35 (1998), 123-143, doi:10.1090/S0273-0979-98-00745-9.

and go to example 6.5 please.

In this article Morel writes that the Rost-Motive of a n-fold Pfister quadric $M_\alpha$ = $M(spec(L))$ for n=1 and for n=2 it coincides with the motive of a conic. He then writes that for n > 2 this is no longer true, meanig that the Rost-Motive is not the motive of a algebraic variety (as far as i understand him).

Where can i find an example for this? Is there some general reason why this happens?

Thank you.

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  • $\begingroup$ Usually Rost motives are direct summands of motives of algebraic varieties, i.e. you want to consider specific pieces of motives of (certain) varieties. $\endgroup$
    – Mikhail Bondarko
    Commented Jan 7, 2014 at 4:42
  • $\begingroup$ Ok i understand this so far.But let me be bit more precise.For a 2 fold,anisotropic Pfister-quadric $X_\phi, \phi = <<a,b>>$ ,the motivic decomposition is $M(X_\phi) = M_\alpha \oplus M_\alpha[1]$. So this contains only one Motive up to twists. As far as i understand the situation,you are trying to tell me that for a n-fold,n>2 Pfister-quadric there will be completely different summands in the decomposition even modulo Tate-twists? This is what i mean. $\endgroup$
    – Jason Pioneer
    Commented Jan 7, 2014 at 18:23
  • $\begingroup$ For any $n$ the Chow motive of $n$-fold Pfister quadric decomposes into Tate twists of the Rost motive. $\endgroup$
    – Victor Petrov
    Commented Apr 3, 2014 at 18:58
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    $\begingroup$ Fixed your reference so that a) people can see what article it is without clicking and b) they aren't directed straight to the pdf. :-) $\endgroup$
    – David Roberts ♦
    Commented Jul 11, 2015 at 16:46

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Over a separable closure the Rost motive becomes isomorphic to à direct sum of 2 tate motives, while the motive of a projective quadric is over a separably closed field either isomorphic to the motive of a projective space (odd dimension) or the motive of a projective space plus another Tate motive. Consequently the Rost motive can only be the motive of the whole quadric in very low dimension. All this is computed in the works of Rost and Vishik.

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