Computational techniques for sheaf cohomology of locally profinite sets

We compute the sheaf cohomology with constant $\mathbb{Z}_2$ coefficients of a concrete class of locally profinite sets of independent interest. We introduce $k$-Fubini partitions to aid in constructions, which witness a failure of a Fubini theorem analog for these spaces. It is also shown that questions of intermediate cohomology degrees can be reduced to questions about top cohomology degrees by exhibiting nontrivial top cocycles as pointwise limits of coboundaries.