Nonabelian cohomology and finiteness theorems for integral orbits of affine group schemes

This paper develops techniques for the nonabelian cohomology $H^1(M,G)$ of a group scheme $M$ finite over a ring $A$ with values in an $A$-group $G$ on which $M$ acts. The finiteness of $H^1(M,G)$ is proved in the case when $A$ is a field of type $(F)$ or a ring of arithmetic type. From this result finiteness theorems are deduced for the decomposition of a $G(A)$ conjugacy class under intersection with the subgroup $G^M(A)$ of fixed integral points of $M$ in $G$ and the more general $G(A)$-orbits.
Bibliography: 20 titles.