Homological finiteness properties of pro-p modules over metabelian pro-p groups

We characterize the modules B of homological type FPm over , where G is a topologically finitely generated metabelian pro-p group that is an extension of A by Q, with A and Q abelian, and B is a finitely generated pro-p -module that is viewed as a pro-p -module via the projection G→Q. The characterization is given in terms of the invariant introduced by King [J.D. King, A geometric invariant for metabelian pro-p groups, J. London Math. Soc. (2) 60 (1) (1999) 83–94] and is a generalization of the case when is considered as a trivial -module that gives the classification of metabelian pro-p groups of type FPm [D.H. Kochloukova, Metabelian pro-p groups of type FPm, J. Group Theory 3 (4) (2000) 419–431].