Cohomology for Frobenius kernels of $S{L}_2$

Let $(S{L}_2{)}_r$ be the $r$-th Frobenius kernels of the group scheme $S{L}_2$ defined over an algebraically field of characteristic $p>2$. In this paper we give for $r\ge 1$ a complete description of the cohomology groups for $(S{L}_2{)}_r$. We also prove that the reduced cohomology ring \opH^\bullet((SL_2)_r,k)_{\red} is Cohen-Macaulay. Geometrically, we show for each $r\ge 1$ that the maximal ideal spectrum of the cohomology ring for $(S{L}_2{)}_r$ is homeomorphic to the fiber product $G{\times }_B{\text{fraku}}^r$. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius-Luzstig kernels of quantized enveloping algebras of type $S{L}_2$.