  We produce a criterion for open sets in projective $n$-space over a separably
closed field to have \'etale cohomological dimension bounded by $2n-3$. We use
the criterion to exhibit a scheme for which \'etale cohomological dimension is
smaller than what a conjecture of G.~Lyubeznik predicts; the discrepancy is of
arithmetic nature.
  For a monomial ideal, we relate extremal graded Betti numbers and \'etale
cohomological dimension of the complement of the corresponding subspace
arrangement. Moreover, we derive upper bounds for its arithmetic rank in terms
of invariants distilled from the lcm-lattice.
