  We study the exactness of certain combinatorially defined complexes which
generalize the Orlik-Solomon algebra of a geometric lattice. The main results
pertain to complex reflection arrangements and their restrictions. In
particular, we consider the corresponding relation complexes and give a simple
proof of the $n$-formality of these hyperplane arrangements. As an application,
we are able to bound the Castelnouvo-Mumford regularity of certain modules over
polynomial rings associated to Coxeter arrangements (real reflection
arrangements) and their restrictions. The modules in question are defined using
the relation complex of the Coxeter arrangement and fiber polytopes of the dual
Coxeter zonotope. They generalize the algebra of piecewise polynomial functions
on the original arrangement.
