On certain representations of automorphism groups of an algebraically closed field

Let k be an algebraically closed field of characteristic zero, F its algebraically closed extension, and G be the group of k-automorphisms of F endowed with a natural topology. One of the purposes of this paper is to show that any non-faithful continuous representation of G factors through a discrete quotient of G. Properties of representation of G arising from geometry are studied. In some cases the groups of morphisms between geometric objects are identified with the groups of morphisms between corresponding G-modules, and the ${\rm Ext}^1$'s are related. In particular, the category of abelian varieties over k with morphisms tensored with the rationals can be described as a category of G-modules.