Low Degree Cohomology of Frobenius Kernels

Let G be a simple algebraic group defined over an algebraically closed field of characteristic \(p>0\). For a positive integer r, let \(G_r\) be the r-th Frobenius kernel of G. We determine in this paper a number m such that the cohomology \(\text {H}^n(G_r,k)\) is isomorphic to \(\text {H}^n(G_1,k)\) for all \(n\le m\) where m depends on p and the type of G.