# Rost-Motive for n > 2

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.

• Usually Rost motives are direct summands of motives of algebraic varieties, i.e. you want to consider specific pieces of motives of (certain) varieties. Commented Jan 7, 2014 at 4:42
• 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. Commented Jan 7, 2014 at 18:23
• For any $n$ the Chow motive of $n$-fold Pfister quadric decomposes into Tate twists of the Rost motive. Commented Apr 3, 2014 at 18:58
• 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. :-) Commented Jul 11, 2015 at 16:46