  Let $G$ be the linear algebraic group $SL_3$ over a field $k$ of
characteristic two. Let $A$ be a finitely generated commutative $k$-algebra on
which $G$ acts rationally by $k$-algebra automorphisms. We show that the full
cohomology ring $H^*(G,A)$ is finitely generated. This extends the finite
generation property of the ring of invariants $A^G$. We discuss where the
problem stands for other geometrically reductive group schemes.
