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

Abstract

This is the first article of a series of two, aiming at constructing and studying motivic Galois groups in the context of triangulated motives. We first develop a general formalism that allows us to associate to a monoidal functor f, satisfying some natural conditions, a Hopf algebra in the target category of f. This formalism is then applied to the Betti realization of Morel-Voevodsky motives over a base field k endowed with a complex embedding σ: k → ℂ. This gives a Hopf algebra Hmot(k, σ) in the derived category of ℚ-vector spaces. Using the comparison theorem between singular and de Rham cohomology, we obtain an explicit description of unitary algebra Hmot(k, sigma;) ℂ showing in particular that the complex Hmot(k, σ) has no homology in strictly negative degrees. We deduce from this a structure of a Hopf algebra on the zeroth homology of Hmot(k, σ) whose spectrum will be called the motivic Galois group.