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Given a separably closed field
and finite degree of imperfection, we study the
functor which takes a semiabelian variety
to the maximal divisible subgroup of
. Our main result is an example where
, as a ‘type-definable group’ in
, does not have ‘relative Morley rank’, yielding a counterexample to a claim in Hrushovski [J. Amer. Math. Soc.9 (1996), 667–690]. Our methods involve studying the question of the preservation of exact sequences by the
functor, and relating this to issues of descent as well as model-theoretic properties of
. We mention some characteristic 0 analogues of these ‘exactness-descent’ results, where differential algebraic methods are more prominent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hasse field, which is interesting in its own right, as well as providing a uniform treatment of the characteristic 0 and characteristic
cases of ‘exactness descent’.
Dichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.
It is known that in differentially closed fields of characteristic zero, the ranks of stability RU, RM and the topological rank RH need not to be equal. Pillay and Pong have just shown however that the ranks RU and RM coincide in a group definable in this theory. At the opposite, we will show in this paper that the ranks RM and RH of a definable group can also be different, and even lead to non-equivalent notions of generic type.
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