On the topology of geometric and rational orbits for algebraic group actions over valued fields, II

Abstract

The aim of this paper is twofold. First, we show that if G is a smooth nilpotent group acting on an algebraic variety V defined over an admissible valued field k and then the Zariski closedness of the geometric orbit in is equivalent to the Hausdorff closedness of the rational orbit in V(k). Second, we provide some calculations for the fact that there is a bijection between the set of G(k)-orbits and the kernel of the natural map in flat cohomology. These results are obtained in the framework of studying the rational orbits.