  We establish a formula which decomposes the cohomologies of various sheaves
on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial
cohomologies of the dual complex $\Delta(D)$ with coefficients in a presheaf of
vector spaces. This presheaf consists precisely of the corresponding cohomology
data on the components of $D$ and on their intersections. We use this formula
to give a Hodge decomposition for SNC divisors and investigate the toric
setting. We also conjecture the existence of such a formula for effective
non-reduced divisors with SNC support, and show that this would imply the
vanishing of the higher simplicial cohomologies of the dual complex associated
to a resolution of an isolated rational singularity.
