0 M ay 2 00 6 REPRESENTATIONS OF FIELD AUTOMORPHISM GROUPS

This is a common introduction to math.AG/0011176, math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. In these papers one studies the automorphism group G of an extension F/k of algebraically closed fields, especially in the case of countable transcendence degree and characteristic zero, its smooth linear and semi-linear representations, and their relations to algebraic geometry (birational geometry, motives, differential forms and sheaves). Compared to the above references there are some new results including • a description of a separable closure of an extension of transcendence degree one of an algebraically closed field (Proposition A.1, p. 11); • a " Künneth formula " for the products with curves; • the semi-simplicity of the G-module Ω n F /k,reg of regular differential forms of top degree. The study of field automorphism groups is an old subject. Without any attempt of describing its complicated history, let me just mention that many topological groups are field automorphism groups. Besides the usual Galois groups we meet here (discrete, p-adic for p < ∞, or finite adelic) groups of points of algebraic groups. Let F/k be a field extension of countable (this will be the principal case) or finite transcendence generalizing the case [K1] of algebraic extension), consider G as a topological group with the base of open subgroups given by the stabilizers of finite subsets of F. Then G is a totally disconnected Hausdorff group, and for any intermediate subfield k ⊆ K ⊆ F the topology on G F/K coincides with the restriction of the topology on G. There are maps: from the set of intermediate subfields in F/k to the set of closed subgroups of G, K → G F/K := Aut(F/K), and from the set of closed subgroups in G to the set of intermediate subfields in F/k, H → F H. They are mutually inverse to each other in the Galois extension case. If n < ∞ then G is locally compact. Following the very general idea, not only in Mathematics, that a " sufficiently symmetric " system is determined by a representation of its symmetry group, one tries to compare various " geometric categories over k " with various categories of representations of G, To ensure that the representation theory of G is rich enough, F should be " big enough " , e.g. algebraically closed. So F is " the function field of the universal tower of n-dimensional k-varieties " , if n …