Modules de cycles et classes non ramifi\'ees sur un espace classifiant

Let G be a finite group of exponent m and let k be a field of characteristic prime to m, containing the m-th roots of unity. For any Rost cycle module M over k, we construct exact sequences detecting the unramified elements in Serre's group of invariants of G with values in M in terms of "residue" morphisms associated to pairs (D,g), where D runs through the subgroups of G and g runs through the homomorphisms \mu_m \to G whose image centralises D. This allows us to recover results of Bogomolov and Peyre on the unramified cohomology of fields of invariants of G, and to generalise them.