Homological determination of $\Gamma$-modules

It is proved that if $M$ is a profinitely generated $\Gamma$-module which is free as a module over the ring of $p$-adic integers, then $M$ is determined up to free direct factors by its homology. This result generalizes the theorem on homological determinacy of $p$-adic representations of a cyclic group [Ref. Zh. Mat., 3A, 318 (1971)].