Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences

Given a separably closed field K of positive characteristic and finite degree of imperfection we study the # functor which takes a semiabelian variety G over K to the maximal divisible subgroup #G of G(K). We show that the # functor need not preserve exact sequences. The main result is an example where #G does not have "relative Morley rank", yielding a counterexample to a claim of Hrushovski. The methods involve studying preservation of exact sequences by the # functor as well as issues of descent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, as well as giving characteristic 0 versions of our results.