  We use a Mayer-Vietoris-like spectral sequence to establish vanishing results
for the cohomology of complements of linear and elliptic hyperplane
arrangements, as part of a more general framework involving duality and abelian
duality properties of spaces and groups. In the process, we consider cohomology
of local systems with a general, Cohen-Macaulay-type condition. As a result, we
recover known vanishing theorems for rank-1 local systems as well as group ring
coefficients, and obtain new generalizations.
