Permutation modules and Chow motives of geometrically rational surfaces

We prove that the Chow motive with integral coefficient of a geometrically rational surfaces S over a perfect field k   is zero dimensional if and only if the Picard group of k¯×kS, where k¯ is an algebraic closure of k  , is a direct summand of a Gal(k¯/k)-permutation module, and S possesses a zero cycle of degree one. As shown by Colliot-Thélène in a letter to the author (which we have reproduced in the appendix) this is in turn equivalent to S   having a zero cycle of degree 1 and CH0(k(S)×kS)CH0(k(S)×kS) being torsion free.