### Construction inconditionnelle de groupes de Galois motiviques, by Yves André and Bruno Kahn

We attach to any ``classical"
Weil cohomology theory H over a field k a motivic Galois group G_H,
well-defined up to inner automorphism. The method is to use results of
earlier work of the two authors [1] to prove that the symmetric monoidal
category Mot^*_H of
Q-linear motives modulo H-equivalence whose K\"unneth projectors are
all algebraic is ``semi-primary" in the sense of loc. cit., that it
fully embeds as a thick subcategory Mot^*_\num into the category
Mot_\num of motives modulo numerical equivalence (which is semi-simple
by Jannsen), and then to apply the main result of [1] to deduce the
existence of monoidal sections s:Mot_\num^*\to Mot_H^*, which are
unique up to isomorphism.

We also study the specialisation of numerical motives and the
behaviour of motivic Galois group by specialisation. If K is a complete
discrete valuation field with residue field k, choose a pair of
compatible classical Weil cohomologies (H_K,H_k). Let
Mot_{H_K,b}^* be the full subcategory of Mot^*_{H_K} consisting of
motives with good reduction, Mot_{\num,b}^* its full image in
Mot_\num^*, and G_{H_K}^b the quotient of G_{H_K} corresponding to
Mot_{\num,b}^*. Then we construct a ``cospecialisation" homomorphism
G_{H_k}\to G_{H_K}^b, compatible with the specialisation of
correspondences and well-defined up to inner automorphism.

[1] Yves André and Bruno Kahn, Nilpotence, radicaux et structures
monoïdales, Preprint of the Institute of Mathematics of Jussieu 297
(2001); updated version available at
http://www.mathematik.uni-bielefeld.de/LAG/

Yves André <andre@math.jussieu.fr>

Bruno Kahn <kahn@math.jussieu.fr>