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 mono´dales, Preprint of the Institute of Mathematics of Jussieu 297 (2001); updated version available at

Yves André <>
Bruno Kahn <>