We give a reduction of the function field Mordell-Lang conjecture to the function field Manin-Mumford conjecture, for abelian varieties, in all characteristics, via model theory, but avoiding recourse to the dichotomy theorems for (generalized) Zariski geometries. Additional ingredients include the "Theorem of the Kernel", and a result of Wagner on commutative groups of finite Morley rank without proper infinite definable subgroups. In positive characteristic, where the main interest lies, there is one more crucial ingredient: "quantifier-elimination" for the corresponding A = p ∞ A(U ) where U is a saturated separably closed field.

Given a separably closed field K of characteristic p > 0 and finite degree of imperfection we study the ♯-functor which takes a semiabelian variety G over K to the maximal divisible subgroup of G(K). Our main result is an example where G ♯ , as a "type-definable group" in K, does not have "relative Morley rank", yielding a counterexample to a claim in [Hr]. 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 G ♯ . We mention some characteristic 0 analogues of these "exactness-descent" * Supported by a Post Doctoral position of the Marie Curie European Network MRTN CT-2004-512234 (MODNET) (Leeds, 2007(Leeds, -2008.† Partially supported by ANR MODIG (ANR-09-BLAN-0047) Model theory and Interactions with Geometry ‡ Supported by a Marie Curie Excellence Chair 024052, and EPSRC grants EP/F009712/1 and EP/I002294/1. 1 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 p cases of "exactness-descent".