Cohomology for Frobenius kernels of SL2SL2

Let (SL2)r(SL2)r be the r  -th Frobenius kernels of the group scheme SL2SL2 defined over an algebraically closed field of characteristic p>0p>0. In this paper we give for r⩾1r⩾1 a complete description of the cohomology groups for (SL2)r(SL2)r. We also prove that the reduced cohomology ring H•((SL2)r,k)redH•((SL2)r,k)red is Cohen–Macaulay. Geometrically, we show for each r⩾1r⩾1 that the maximal ideal spectrum of the cohomology ring for (SL2)r(SL2)r is homeomorphic to the fiber product G×BurG×Bur. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius–Lusztig kernels of quantized enveloping algebras of type SL2SL2.