Generalized Cocycles with Values in One-Units of Henselian Valued Division Algebras

Let Z/F be an inertial Galois extension of Henselian valued fields, and let D be a Z-central division algebra. Let G be a finite group acting on Z with fixed field F. We show that every generalized cocycle of G with values in the one-units of D is cohomologous to one of the form (θ, 1), or in other words, the existence of such a cocycle implies that the group action of G on Z extends to a group action on D. We provide applications to lifting of group actions on the residue division algebra and to the existence of Kummer subfields in D given suitable data in [formula].