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

We characterize the modules B of homological type FPm over Z(p) [G], 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 Z(p) [Q]-module that is viewed as a pro-p Z(p) [G]-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 B = Z(p) is considered as a trivial Z(p) [G]-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]. (c) 2005 Published by Elsevier Inc.