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

TL;DR

This paper investigates the properties of the maximal divisible subgroup functor on semiabelian varieties over separably closed fields, providing counterexamples to previous assumptions and developing new structural notions.

Contribution

It demonstrates that the maximal divisible subgroup functor does not always preserve exact sequences and introduces the concept of iterative D-structures on group schemes.

Findings

Counterexample to the preservation of exact sequences by the # functor

Existence of semiabelian varieties with # lacking relative Morley rank

Development of iterative D-structures on group schemes

Abstract

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.