  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, where this homology
was computed for the natural sheaf, itself 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.
