Let \({\mathcal G}\) be a simple, simply connected algebraic group defined and split over the finite field of p elements, let G be the points of \({\mathcal G}\) in an algebraically closed field k and \(G_ 1\) the scheme theoretic kernel of the Frobenius morphism from G to itself. One must assume that p is sufficiently big, depending on the Coxeter diagram of G. Earlier work of the authors showed how to connect the cohomology of G with the cohomology of \(G_ 1\). This paper shows that \(H^*(G_ 1,k)\) is isomorphic as G-module to the coordinate ring \(A^*\) of the variety of nilpotent elements in Lie(G) twisted once by Frobenius. Also that \(A^*\) is acyclic in a certain sense. This makes certain spectral sequences simplify, providing very good information about the cohomology of G with coefficients in rational G-modules.