On unipotent algebraic \(G\)-groups and 1-cohomology. (Q2855931)

Let \(G\) be a connected linear algebraic group over an algebraically closed field \(k\) and \(Q\) a connected unipotent \(G\)-group. It is shown that \(Q\) has a central series \(Q=Q_1\geq Q_2\cdots\) whose layers are \(G\)-modules and such that, as a variety, \(Q\) is the product of the layers. This is derived from a result of Rosenlicht. Versions of the `five lemma' are given involving the non-Abelian cohomology groups \(H^1(G,Q_i)\), \(H^1(G,Q_i/Q_{i+1})\). Now let \(G\) be reductive. The central series \(\{Q_i\}\) is used to extend known results for cohomology of \(G\)-modules to the non-Abelian cohomology groups of type \(H^1(G,Q)\). For instance, if \(P\) is a parabolic subgroup of \(G\), then it is shown that \(H^1(G,Q)=H^1(P,Q)\). If \(G\) is defined over the prime field, one has a generic cohomology group \(H^1_{gen}(G,Q)\) with the usual stability properties. The author uses these results to obtain corollaries about \(G\)-complete reducibility and subgroup structure.