Cohomology for Frobenius kernels of $SL_2$

Let (SL2)r be the r-th Frobenius kernels of the group scheme SL2 defined over an algebraically closed field of characteristic p>0. In this paper we give for r⩾1 a complete description of the cohomology groups for (SL2)r. We also prove that the reduced cohomology ring H•((SL2)r,k)red is Cohen–Macaulay. Geometrically, we show for each r⩾1 that the maximal ideal spectrum of the cohomology ring for (SL2)r is homeomorphic to the fiber product G×Bur. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius–Lusztig kernels of quantized enveloping algebras of type SL2.