We construct minimal cellular resolutions of squarefree monomial ideals
arising from hyperplane arrangements, matroids and oriented matroids.
These are Stanley-Reisner ideals of complexes of independent sets,
and of triangulations of Lawrence matroid polytopes.
Our resolution provides a cellular realization of Stanley's formula
for their Betti numbers. For unimodular matroids our resolutions are
related to hyperplane arrangements on tori, and we recover the resolutions
constructed by D. Bayer, S. Popescu, and B. Sturmfels [BPS].
We resolve the combinatorial problems posed in [BPS]
by computing Möbius invariants
of graphic and cographic arrangements in terms of