Bounding the Dimensions of Rational Cohomology Groups

Let k be an algebraically closed field of characteristic p > 0, and let G be a simple simply-connected algebraic group over k. In this paper we investigate situations where the dimension of a rational cohomology group for G can be bounded by a constant times the dimension of the coefficient module. As an application, effective bounds on the first cohomology of the symmetric group are obtained. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.