Motivic cohomology and infinitesimal group schemes

For $k$ a perfect field of characteristic $p>0$ and $G/k$ split reductive group with $p$ non-torsion prime for $G,$ we compute the mod motivic cohomology geometric classifying space $BG_{(r)}$, where $G_{(r)}$ is $r$th Frobenius kernel $G.$ Our main tool version Eilenberg-Moore spectral sequence, due to Krishna. flat affine scheme finite type, define cycle class map from $BG$ \'etale stack $\mathcal{B}G.$ This also gives into Hodge We study some examples, including kernels.