For any finite simplicial complex
,
Davis and Januszkiewicz defined a family of homotopy equivalent CW–complexes
whose integral cohomology rings are isomorphic to the Stanley–Reisner algebra of
.
Subsequently, Buchstaber and Panov gave an alternative construction, which they
showed to be homotopy equivalent to the original examples. It is therefore
natural to investigate the extent to which the homotopy type of a space
is determined by such a cohomology ring. Having analysed this problem
rationally in Part I, we here consider it prime by prime, and utilise Lannes’
–functor
and Bousfield–Kan type obstruction theory to study the
–completion of
. We find the
situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion
whenever
is a join of skeleta of simplices. We apply our results to the global problem by
appealing to Sullivan’s arithmetic square, and deduce integral uniqueness whenever
the Stanley–Reisner algebra is a complete intersection.