  The purpose of this paper is to exhibit fine structure for polyhedral
products Z(K;(X,A) and polyhedral smash products $\widehat{Z}(K;(X,A)$.
(Moment-angle complexes are special cases for which (X,A) = (D^2,S^1)). There
are three main parts. The first defines a natural filtration of the polyhedral
product and derives properties of the resulting spectral sequence. This is
followed with applications. The second part uses the first to give a
homological decomposition of the polyhedral smash product. Finally there are
applications to the ring structure of H*(Z(K;(X,A))) for CW-pairs (X,A)
satisfying suitable freeness conditions.
