Computing the Cech cohomology of decomposition spaces

A line pattern in a free group $F$ is defined by a malnormal collection of cyclic subgroups. Otal defined a decomposition space $\mathcal{D}$ associated to a line pattern. We provide an algorithm that computes a presentation for the \v{C}ech cohomology of $\mathcal{D}$, thought of as a $F$-module. This answers a relative version of a question of Epstein about boundaries of hyperbolic groups.