Poly-adic filtrations, standardness, complementability and maximality

Given some essentially separable filtration (Zn)n.0 indexed by the nonpositive integers, we define the notion of complementability for the filtrations contained in (Zn)n.0. We also define and characterize the notion of maximality for the poly-adic sub-filtrations of (Zn)n.0. We show that any poly-adic sub-filtration of (Zn)n.0 which can be complemented by a Kolmogorovian filtration is maximal in (Zn)n.0. We show that the converse is false, and we prove a partial converse, which generalizes Vershik.s lacunary isomorphism theorem for poly-adic filtrations.