Detecting Rational Cohomology of Algebraic Groups

Let G be a connected, semisimple algebraic group defined over an algebraically closed field k of positive characteristic p. Assume that G is defined and split over the prime field k0 = GF (p), and for q = p , let G(q) be the subgroup of GF (g)-rational points. Let V be a rational G-module, and, for a non-negative integer r, let V(r) be the rational G-module obtained by 'twisting' the original G-action on V by the r-th power of the Frobenius endomorphism a of G. In [5] we showed (together with W. van der Kallen) that, if V is finite dimensional and n is a non-negative integer, for sufficiently large m and r (depending on V and n), there are isomorphisms