Galois theory, automorphism groups of prime models, and the Picard-Vessiot closure

The paper has three parts or aspects. The first belongs to “pure” model theory. We work in the context of a complete totally transcendental theory T = T e q . We consider the prime model M A over a set A . For intermediate sets B with A ⊆ B ⊆ M A which are normal ( A u t ( M A / A ) -invariant) and “minimal” we give a full Galois correspondence between intermediate definably closed sets A ⊆ B ⊆ M A and relatively type-definable subgroups of A u t ( B / A ) (the group of A -elementary permutations of B ). The unique greatest such minimal normal B coincides with Poizat's “minimal closure” A m i n . Our paper can be see as extending the well-known Galois correspondence between closed subgroups of the profinite group A u t ( a c l ( A ) / A ) and intermediate definably closed sets, to the case of A m i n in place of a c l ( A ) The second aspect is an application of the above to the “Picard-Vessiot closure” K P V ∞ (or K ∞ ) of a differential field K of characteristic 0 with algebraically closed field C K of constants (where the ambient first order theory is D C F 0 ). We also show that normal differential subfields of K P V ∞ containing K are “iterated PV -extensions” of K , and the Galois correspondence result above holds for these extensions. This fills in some missing parts of Magid's paper [6] . The third aspect is a study of exact sequences 1 → N → G → H → 1 , where G = A u t ( K 2 / K ) , N = A u t ( K 2 / K 1 ) and H = A u t ( K 1 / K ) , K 1 is a (maybe infinite type) PV extension of K , K 2 is a (maybe infinite type) PV extension of K 1 and K 2 is normal over K (in the differential closure of K ) and again C K is algebraically closed. Both N and H have the structures of proalgebraic groups over C K . We show that conjugation by any given element of G is a proalgebraic automorphism of N . Moreover if G splits as a semidirect product N ⋊ H , then left multiplication by any fixed element of G is a morphism of proalgebraic varieties N × H → N × H . This improves and extends observations in Section 4 of [6] which dealt with one particular example.