  A spectral sequence is defined which converges to the \v{C}ech cohomology of
the Euclidean hull of a tiling of the plane with Euclidean finite local
complexity. The terms of the second page are determined by the so-called ePE
homology and ePE cohomology groups of the tiling, and the only potentially
non-trivial boundary map has a simple combinatorial description in terms of its
local patches. Using this spectral sequence, we compute the \v{C}ech cohomology
of the Euclidean hull of the Penrose tilings.
