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The maximal divisible type-definable subgroup $G^\# = [p^\infty]G(K) =
\bigcap_n [p^n]G(K)$ of a semiabelian variety over a separably closed field
$K$ played a crucial role in Hrushovski's celebrated proof of function-field
Mordell-Lang in positive characteristic, the crucial idea being that $G^\#$ is
the positive characteristic analogue of the ``Manin kernel'' in characteristic
zero (also denoted $G^\#$).

This paper treats in detail the behavior of the $\cdot^\#$ functor (in all
characteristics) when applied to short exact sequences of semiabelian
varieties, and finds that the question of when it preserves exactness is
closely linked to descent: if $0\rightarrow G_1\rightarrow G_2 \rightarrow G_3
\rightarrow 0$ is a (separable) exact sequence of ordinary semiabelian
varieties and $G_1$ and $G_3$ are defined over the absolute constants, then
$G_2$ descends to the absolute constants iff $0\rightarrow G_1^\# \rightarrow
G_2^\# \rightarrow G_3^\# \rightarrow 0$ is also exact.

Using this equivalence, the authors find examples in all characteristics of
semiabelian varieties $0\rightarrow \mathbb{G}_m^n \rightarrow S \rightarrow
A\rightarrow 0$ where the $\cdot^\#$ functor is not exact. In positive
characteristic, they find that in this case $S^{\#}$ does not have
ordinal-valued ``relative Morley rank'' (being the analogue of Morley rank for
relatively definable subsets of a type-definable set). The existence of such
$S$ contradicts a claim in \cite{HrML}. The present paper includes in the
introduction a sketch of how to avoid using that claim in the relevant part of
the argument of \cite{HrML}, so the validity of \cite{HrML} is unaffected.

The authors also take the opportunity to discuss some of the basic properties
of relative Morley rank. This is a very natural notion, which appeared in a
less general form in \cite{HrML}, and is of interest in itself.

The results on exactness and descent are proven in arbitrary characteristic.
This proceeds by exploiting the same correspondence between separably closed
fields and differentially closed fields, and between $[p^\infty]G(K)$ and the
Manin kernel, exploited in Hrushovski's work. The authors formalise this
correspondence by developing the theory of ``D-structures'' in arbitrary
characteristic, generalising the theory of Buium in characteristic zero. In
particular, they find that taking the inverse limit of copies of $G$ under
$[p^n]$ as the positive characteristic analogue of the universal vectorial
extension in characteristic zero, applying the characteristic zero definition
of $G^{\#}$ in positive characteristic does indeed yield $[p^\infty]G(K)$.

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