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" $K^{PV_{\infty}}$ of a differential field $K$ of char $0$ with algebraically closed field $C_{K}$ of constants. We also show that normal differential subfields of $K^{PV_{\infty}}$ 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 $N\rtimes H$, 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.