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

Abstract

We work in the context of a complete totally transcendental theory $T=T^{eq}$. We consider the prime model $M_A$ over a set $A$. For intermediate sets $B$ with $A\subseteq B\subseteq M_A$ which are normal ($Aut(M_A/A)$-invariant) and ``minimal" we give a full Galois correspondence between intermediate definably closed sets $A\subseteq B\subseteq M_A$ and ``closed" subgroups of $Aut(B/A)$ (the group of $A$-elementary permutations of $B$). The unique greatest such minimal normal $B$ coincides with Poizat's ``minimal closure" $A_{min}$, so our paper extends (from $acl(A)$ to $A_{min}$) the well-known Galois correspondence between closed subgroups of the profinite group $Aut(acl(A)/A)$ and intermediate definably closed sets. The main result applies to the ``Picard-Vessiot closure" $K^{P{V}_{\infty }}$ of a differential field $K$ of char $0$ with algebraically closed field $C_K$ of constants. We also show that normal differential subfields of $K^{P{V}_{\infty }}$ containing $K$ are ``iterated $PV$-extensions" of $K$, and the Galois correspondence above holds for these extensions. This fills in some missing parts of Magid's paper [5]. We also discuss exact sequences $1\to N\to G\to H\to 1$, where $G=Aut(K_2/K)$, $N=Aut(K_2/K_1)$ and $H=Aut(K_1/K)$, $K_1$ 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 $K$ and again $C_K$ is algebraically closed. Both $N$ and $H$ have the structure 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 $N\times H\to N\times H$. This improves and extends observations in Section 4 of [5] which dealt with one example.

21 pages, edits to the introduction