#### ABSTRACT:

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 $$\mathbb C\{x\}$$ can always be characterized in terms of $$v$$-curvatures, in the spirit of the work of Katz (1982) [8]. 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.

HAL:hal-00629578

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