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We study the exactness of certain combinatorially defined complexes whichgeneralize the Orlik-Solomon algebra of a geometric lattice. The main resultspertain to complex reflection arrangements and their restrictions. Inparticular, we consider the corresponding relation complexes and give a simpleproof of the $n$-formality of these hyperplane arrangements. As an application,we are able to bound the Castelnouvo-Mumford regularity of certain modules overpolynomial rings associated to Coxeter arrangements (real reflectionarrangements) and their restrictions. The modules in question are defined usingthe relation complex of the Coxeter arrangement and fiber polytopes of the dualCoxeter zonotope. They generalize the algebra of piecewise polynomial functionson the original arrangement.

Relation spaces of hyperplane arrangements and modules defined by graphs of fiber zonotopes