Motives and admissible representations of automorphism groups of fields

Abstract

Some of basic properties of the groups of automorphisms of algebraically closed fields and of their smooth representations are studied. In characteristic zero, Grothendieck motives modulo numerical equivalence are identified with a full subcategory in the category of graded smooth representations of certain automorphism groups of algebraically closed fields.

Semilinear representations of PGL

“…The paper is motivated by a study of admissible representations 2 of the automorphism group G of an algebraically closed extension F of k of countable transcendence degree undertaken in [4]. One can expect that any such representation is contained in an appropriate admissible semilinear representation (cf.…”

“…This is a topological group with the base of open subgroups formed by the stabilizers of finite subsets in F ; more details can be found in [4]. Vol.…”

“…If we could show that G F/ξ(L 1 ) ⊂ G ′ for any ξ ∈ G F/L 0 then G ′ would contain an open normal subgroup in G F/L 0 . It follows from the simplicity of G F/L 0 (Theorem 2.9 of [4]) that G ′ = G F/L 0 . Let x 1 , .…”

“…By Lemma 6.1 of [4], V G F /L ∞ = L⊂L ∞ V G F /L , where L runs over subfields of finite type over k. Then such L are contained in appropriate L j , so…”

“…The paper is motivated by a study of admissible representations of the automorphism group G of an algebraically closed extension of k of countable transcendence degree undertaken in [4]. The semigroup End(L/k) is considered as a subquotient of G, hence the condition on extendability.…”