Sheaf homology of hyperplane arrangements, Boolean covers and exterior powers

We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coefficients in the graded exterior sheaf of the natural sheaf. This builds on the results of our previous paper [EverittTurner19a], which in turn are a generalisation of an old result of Lusztig. The computational machinery we develop in this paper is quite different though: sheaf homology is lifted to what we call Boolean covers, where we instead compute homology cellularly. A number of tools are given for the cellular homology of these Boolean covers, including a deletion-restriction long exact sequence.