Rost projectors and Steenrod operations

Abstract

Let $X$ be an anisotropic projective quadric possessing a Rost projector $\rho$. We compute the 0-dimensional component of the total Steenrod operation on the modulo 2 Chow group of the Rost motive given by the projector $\rho$. The computation allows to determine the whole Chow group of the Rost motive and the Chow group of every excellent quadric (the results announced by Rost). On the other hand, the computation is being applied to give a simpler proof of Vishik's theorem stating that the integer $\dim X+1$ is a power of 2.