  Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$
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
$(SL_2)_r$. 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
$(SL_2)_r$ is homeomorphic to the fiber product $G\times_B\fraku^r$. Finally,
we adapt our calculations to obtain analogous results for the cohomology of
higher Frobenius-Luzstig kernels of quantized enveloping algebras of type
$SL_2$.
