Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences
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.
Keywords
Cite
@article{arxiv.0904.2083,
title = {Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences},
author = {Franck Benoist and Elisabeth Bouscaren and Anand Pillay},
journal= {arXiv preprint arXiv:0904.2083},
year = {2015}
}
Comments
55 pages In this version 3, some corrections and clarifications are made: in section 2.3 on relative Morley rank. Also in section 5.2 where more explanation is given of D-structures in positive characteristic. In an appendix we give a proof of the exactness of the functor taking a semiabelian variety to its universal vectorial extension