On maximal proper subgroups of field automorphism groups

Let $G$ be the automorphism group of an extension $F|k$ of algebraically closed fields of characteristic zero and of transcendence degree $n$, $1\le n\le\infty$. In this paper we (i) construct some maximal closed non-open subgroups $G_v$, and some (all, in the case of countable transcendence degree) maximal open proper subgroups of $G$; (ii) describe, in the case of countable transcendence degree, the automorphism subgroups over the intermediate subfields (a question of Krull, \cite[\S4, question 3b)]{krull}); (iii) construct, in the case $n=\infty$, a fully faithful subfunctor $(-)_v$ of the forgetful functor from the category of smooth representations of $G$ to the category of smooth representations of $G_v$; (iv) construct, using the functors $(-)_v$, a subfunctor $\Gamma$ of the identity functor on the category of smooth representations of $G$, coincident (via the forgetful functor) with the functor $\Gamma$ on the category of smooth admissible semilinear representations of $G$ constructed in \cite{adm} in the case $n=\infty$ and $k=\bar{{\mathbb Q}}$. The study of open subgroups is motivated by the study of (the stabilizers of the) smooth representations undertaken in \cite{repr,adm}. The functor $\Gamma$ is an analogue of the global sections functor on the category of sheaves on a smooth proper algebraic variety. Another result is that `interesting' semilinear representations are `globally generated'.