  A certain Grothendieck topology assigned to a metric space gives rise to a
sheaf cohomology theory which sees the coarse structure of the space. Already
constant coefficients produce interesting cohomology groups. In degree 0 they
see the number of ends of the space. In this paper a resolution of the constant
sheaf via cochains is developed. It serves to be a valuable tool for computing
cohomology. In addition coarse homotopy invariance of coarse cohomology with
constant coefficients is established. This property can be used to compute
cohomology of Riemannian manifolds. The Higson corona of a proper metric space
is shown to reflect sheaves and sheaf cohomology. Thus we can use topological
tools on compact Hausdorff spaces in our computations. In particular if the
asymptotic dimension of a proper metric space is finite then higher cohomology
groups vanish. We compute a few examples. As it turns out finite abelian groups
are best suited as coefficients on finitely generated groups.
