Let $X$ be a stone space, i.e. a compact, totally disconnected hausdorff space. Then $H^1(X,\mathbb{Z}/2)=0$. Here's one way of proving this: $X$ with $\mathbb{Z}/2$ (the constant sheaf) is an affine scheme, now use the vanishing result for quasicoherent sheaves on affine schemes.


What happens if we also allow locally compact, totally disconnected hausdorff spaces? Then $X$ with $\mathbb{Z}/2$ is again a scheme, but it is not affine (unless $X$ is compact). A typical example would be an open subset of a stone space (actually this is generic), or the underlying topological space of a local field.



Sheaf Cohomology on a Stone Space
A search brought up [Sheaf Cohomology of Locally Compact Totally Disconnected Spaces](https://doi.org/10.2307/2035693) by R. Wiegand. [Some topological invariants of Stone spaces](https://doi.org/10.1307/mmj/1029000311) by the same author might also be of interest.


