Abstract

For $k$ a perfect field of characteristic $p>0$ and $G∕k$ a split reductive group with $p$ a nontorsion prime for $G$, we compute the mod $p$ motivic cohomology of the geometric classifying space $B{G}_{(r)}$, where $G_{(r)}$ is the $r$-th Frobenius kernel of $G$. Our main tool is a motivic version of the Eilenberg–Moore spectral sequence, due to Krishna.

For an algebraic group $G∕k$, we define a cycle class map from the mod $p$ motivic cohomology of the classifying space $BG$ to the mod $p$ étale motivic cohomology of the classifying stack $\mathcal{ℬ}G$. This also gives a cycle class map into the Hodge cohomology of $\mathcal{ℬ}G$. We study the cycle class map for some examples, including Frobenius kernels.

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