L'algèbre de Hopf et le groupe de Galois motiviques d'un corps de caractéristique nulle, II

Abstract

This is the second article of a series of two, aiming at constructing and studying motivic Galois groups in the context of triangulated motives. These motivic Galois groups were constructed in the first article and their algebras of regular functions were described concretely in terms of differential forms or algebraic cycles. In the present article, we gather important complements to the first one. In the first part of this article, we describe the link between the motivic Galois group and the usual Galois group of a subfield of ℂ. In the second part, we develop the basis of a theory of ramification for the motivic Galois groups by constructing motivic versions of the decomposition and inertia groups associated to a geometric valuation. We also introduce the relative motivic Galois group of an extension K/k which measures the difference between the motivic Galois groups of k and K. The latter is more accessible than its absolute counterpart. In effect, we show that it is a quotient of the pro-algebraic completion of the topological fundamental pro-group of a the pro-variety homk(K, ℂ), at least when the extension K/k is of finite type.