Curvatures, generic Galois groups and D-groupoid of a q-difference system

Combining the results in Hendriks (1996) [6], Di Vizio (2002) [1] and Di Vizio, Hardouin [2], we prove that the generic, algebraic or differential, Galois group of a q-difference modules over C(x) can always be characterized in terms of v-curvatures, in the spirit of the work of Katz (1982) In We use this result to prove that the Malgrange-Granier D-groupoid of a linear q-difference system coincide, in a sense that we specify below, with a sort of Kolchin closure of the dynamics of the linear q-difference system and that the group that fixes a transversal, coincide with the differential generic Galois group. (C) 2010 Academie des sciences. Publie par Elsevier Masson SAS. Thus droits reserves.