We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coefficients in the graded exterior sheaf varLambda ^bullet F of the natural sheaf F. This builds on the results of our previous paper Everitt and Turner (Adv Math 402:Paper No. 108354, 2022. https://doi.org/10.1016/j.aim.2022.108354) where this homology was computed for varLambda ^1F=F, 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.